>Systems of Equations>Solve Systems of Equations with Multiple Unknowns

Solve Systems of Equations with Multiple Unknowns

When you’re solving equations with three variables (or unknowns), you do the same as you did previously with the substitution method. The only difference is that it has one more step.

I recommend that you split your page into three columns, with one equation in each column. Read the instructions below carefully!

Rule

How to solve equations with three variables

1.: Select one of the equations and solve it for one of the variables.
2.: Take the first equation and input it into the other two equations. Arrange them nicely in your columns.
3.: Now you have two equations with two variables. You can solve them both with the substitution method.
4.: Using the two answers you now have, you input them into the equation you started with to find the final variable.

Example 1

Solve the set of equations:

\begin{align} 2 x - 3 y + 4 z = 24 (1) x - 2 y + 5 z & = 25 & (2) 3 x + 5 y + 3 z & = 5 & (3)1. Select (2) and solve for x : \begin{matrix} x - 2 y + 5 z = 25 \\ x = 25 + 2 y - 5 z \end{matrix} \begin{align} x - 2 y + 5 z & = 25 \\ x & = 25 + 2 y - 5 z & (4) \end{align} 2. Substitute your answer into the two other equations and tidy it: (1): \begin{matrix} 2 x - 3 y + 4 z = 24 \\ 2 (25 + 2 y - 5 z) - 3 y + 4 z = 24 \\ 50 + 4 y - 10 z - 3 y + 4 z = 24 \\ y - 6 z = - 26 \end{matrix} \begin{array}{l} 2 x - 3 y + 4 z & = 24 \\ 2 (25 + 2 y - 5 z) - 3 y + 4 z & = 24 \\ 50 + 4 y - 10 z - 3 y + 4 z & = 24 \\ y - 6 z & = - 26 \end{array} This is equal to y = 6 z - 26 . (5) (3): \begin{array}{l} 3 x + 5 y + 3 z & = 5 \\ 3 (25 + 2 y - 5 z) + 5 y + 3 z & = 5 \\ 75 + 6 y - 15 z + 5 y + 3 z & = 5 \end{array} This simplifies to 11 y - 12 z = - 70 (6) 3. Solve the set of equations with the two new equations. Substitute (5) into         (6): \begin{array}{l} 11 y - 12 z & = - 70 \\ 11 (6 z - 26) - 12 z & = - 70 \\ 66 z - 286 - 12 z & = - 70 \\ 54 z & = 216 \\ z & = 4 \end{array} Substitute this back into (5) and you get: y = 6 \cdot 4 - 26 = - 2 . 4. Substitute the two answers into (4) and solve: \begin{array}{l} x & = 25 + 2 y - 5 z \\ = 25 + 2 \cdot (- 2) - 5 \cdot 4 \\ = 25 - 4 - 20 \\ = 1 . \end{array} The final answer is then (x, y, z) = (1, - 2, 4) .Want to know more? Sign Up It's free!Previous entry How to Solve Systems of Nonlinear EquationsGo backHome Curriculum Stats LeagueAI Ask the OracleAI How can I help you? BetaAbout Us About House of Math Employees Career Media Lectures Blog Contact support@houseofmath.com Booking of private tutoring +47 22 150 300 Address Bygdøy allé 23
            0262 Oslo
            Norway FAQ Links WeTakeAction Math Magic for Ukraine Tutoring Bootcamps Junior Math Pen & Paper exercises Encyclopedia Book a Tutor Language/Curriculum Terms Privacy Cookies Download our app{"props":{"pageProps":{"campaign":false,"articleBody":" When you’re solving equations with three variables (or unknowns), you do the same as you did previously with the substitution method. The only difference is that it has one more step. I recommend that you split your page into three columns, with one equation in each column. Read the instructions below carefully! Rule How to solve equations with three variables 1. Select one of the equations and solve it for one of the variables. 2. Take the first equation and input it into the other two equations. Arrange them nicely in your columns. 3. Now you have two equations with two variables. You can solve them both with the substitution method. 4. Using the two answers you now have, you input them into the equation you started with to find the final variable. Example 1 Solve the set of equations: 2x − 3y + 4z = 24 (1) x − 2y + 5z = 25 (2) 3x + 5y + 3z = 5 (3) 1. Select (2) and solve for x: x − 2y + 5z = 25 x = 25 + 2y − 5z x − 2y + 5z = 25 x = 25 + 2y − 5z (4) 2. Substitute your answer into the two other equations and tidy it: (1): 2x − 3y + 4z = 24 2(25 + 2y − 5z) − 3y + 4z = 24 50 + 4y − 10z − 3y + 4z = 24 y − 6z = −26 2x − 3y + 4z = 24 2(25 + 2y − 5z) − 3y + 4z = 24 50 + 4y − 10z − 3y + 4z = 24 y − 6z = −26 This is equal to y = 6z − 26. (5) (3): 3x + 5y + 3z = 5 3(25 + 2y − 5z) + 5y + 3z = 5 75 + 6y − 15z + 5y + 3z = 5 This simplifies to 11y − 12z = −70 (6) 3. Solve the set of equations with the two new equations. Substitute (5) into (6): 11y − 12z = −70 11(6z − 26) − 12z = −70 66z − 286 − 12z = −70 54z = 216 z = 4 Substitute this back into (5) and you get: y = 6 ⋅ 4 − 26 = −2. 4. Substitute the two answers into (4) and solve: x = 25 + 2y − 5z = 25 + 2 ⋅ (−2) − 5 ⋅ 4 = 25 − 4 − 20 = 1. The final answer is then (x,y,z) = (1,−2, 4). ","topic":{"breadcrumbs":[{"_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c","h1":"Encyclopedia","path":"/encyclopedia"},{"_key":"e9327e4b-42fe-571b-bcb9-83fdb63d5f21","h1":"Algebra","path":"/encyclopedia/algebra"},{"_key":"100a0f18-3f76-5653-91ef-a410836c9583","h1":"Equations and Inequalities","path":"/encyclopedia/algebra/equations-and-inequalities"},{"_key":"f99cf246-2b53-56b2-97b1-efb74d01f11e","h1":"Equations","path":"/encyclopedia/algebra/equations-and-inequalities/equations"},{"_key":"41927a7e-eb51-55a1-908a-da0aea694d01","h1":"Systems of Equations","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations"},{"_key":"bf8c4ec5-bebc-53e9-a95c-713550577ce3","h1":"Solve Systems of Equations with Multiple Unknowns","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/solve-systems-of-equations-with-multiple-unknowns"}],"cssUrl":"/b52af2c0-3875-41cc-a042-0495ec22c744-en-1.css","h1":"Solve Systems of Equations with Multiple Unknowns","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     When you’re solving equations with three variables (or unknowns), you do the same as you did previously with \u003ca href=\"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/solve-systems-of-equations-with-the-substitution-method\"\u003ethe substitution method\u003c/a\u003e. The only difference is that it has one more step. \u003c/p\u003e\u003cp class=\"indent\" \u003e     I recommend that you split your page into three columns, with one equation in each column. Read the instructions below carefully! \u003c/p\u003e      \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-618\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Rule\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"rule-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eHow\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eto\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003esolve\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eequations\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ewith\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ethree\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003evariables\u003c/span\u003e \u003c/h3\u003e      \u003c/p\u003e\u003cdl class=\"enumerate-enumitem\"\u003e\u003cdt class=\"enumerate-enumitem\"\u003e    1.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Select         one         of         the         equations         and         solve         it         for         one         of         the         variables.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Take         the         first         equation         and         input         it         into         the         other         two         equations.         Arrange         them         nicely         in         your         columns.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Now         you         have         two         equations         with         two         variables.         You         can         solve         them         both         with         the         substitution         method.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     4.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Using         the         two         answers         you         now         have,         you         input         them         into         the         equation         you         started         with         to         find         the         final         variable.         \u003cp class=\"noindent\" \u003e\u003c/dd\u003e\u003c/dl\u003e \u003c/p\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                                   \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-619\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the set of equations:\u003c/span\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align\"\u003e                                 \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003cmstyle     id=\"x1-42068r1\"  class=\"label\" \u003e\u003c/mstyle\u003e\u003cmstyle  class=\"maketag\"\u003e\u003cmtext  \u003e(1)\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"0.33em\" class=\"nbsp\" /\u003e                                         \u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003cmstyle     id=\"x1-42069r2\"  class=\"label\" \u003e\u003c/mstyle\u003e\u003cmstyle  class=\"maketag\"\u003e\u003cmtext  \u003e(2)\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"0.33em\" class=\"nbsp\" /\u003e                                         \u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                                  \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003cmstyle     id=\"x1-42070r3\"  class=\"label\" \u003e\u003c/mstyle\u003e\u003cmstyle  class=\"maketag\"\u003e\u003cmtext  \u003e(3)\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"0.33em\" class=\"nbsp\" /\u003e \u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                               \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                                   \u003cdl class=\"enumerate-enumitem\"\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     1.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eSelect (\u003ca  href=\"#\"\u003e2\u003c/a\u003e) and solve for \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math type=\"divide\"\u003e:         \u003cdiv class=\"tight\"\u003e         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e \u003cmtable  class=\"gather-star\"\u003e \u003cmtr\u003e  \u003cmtd\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mtd\u003e  \u003cmtd\u003e\u003c/mtd\u003e \u003c/mtr\u003e\u003cmtr\u003e  \u003cmtd\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e  \u003cmtd\u003e\u003cmstyle     id=\"x1-420711\"  class=\"label\" \u003e\u003c/mstyle\u003e\u003c/mtd\u003e                                                                   \u003c/mtr\u003e\u003c/mtable\u003e \u003c/math\u003e         \u003cp class=\"nopar\" \u003e         \u003c/div\u003e \u003cdiv class=\"wide\"\u003e         \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align\"\u003e                                 \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-label\"\u003e                                 \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                                \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003cmstyle     id=\"x1-42072r4\"  class=\"label\" \u003e\u003c/mstyle\u003e\u003cmstyle  class=\"maketag\"\u003e\u003cmtext  \u003e(4)\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"0.33em\" class=\"nbsp\" /\u003e \u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e         \u003cp class=\"noindent\" \u003e\u003c/div\u003e         \u003c/p\u003e\u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eSubstitute your answer into the two other equations and tidy it:         \u003cp class=\"noindent\" \u003e(\u003ca  href=\"#\"\u003e1\u003c/a\u003e):  \u003cdiv class=\"tight\"\u003e         \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e \u003cmtable  class=\"gather-star\"\u003e \u003cmtr\u003e  \u003cmtd\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mtd\u003e                    \u003cmtd\u003e\u003c/mtd\u003e \u003c/mtr\u003e\u003cmtr\u003e  \u003cmtd\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mtd\u003e  \u003cmtd\u003e\u003c/mtd\u003e \u003c/mtr\u003e\u003cmtr\u003e  \u003cmtd\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mtd\u003e    \u003cmtd\u003e\u003c/mtd\u003e \u003c/mtr\u003e\u003cmtr\u003e  \u003cmtd\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mtd\u003e                          \u003cmtd\u003e\u003c/mtd\u003e                                                                   \u003c/mtr\u003e\u003c/mtable\u003e \u003c/math\u003e         \u003cp class=\"nopar\" \u003e         \u003c/div\u003e \u003cdiv class=\"wide\"\u003e         \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                             \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e                             \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e                             \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e                             \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                                                      \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e         \u003cp class=\"noindent\" \u003e\u003c/div\u003e  This is equal to \u003c/p\u003e\u003ctable class=\"equation\"\u003e\u003ctr\u003e\u003ctd\u003e         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                               \u003cmstyle     id=\"x1-42074r5\"  class=\"label\" \u003e\u003c/mstyle\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e \u003c/math\u003e\u003c/td\u003e\u003ctd class=\"eq-no\"\u003e(5)\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e         \u003cp class=\"noindent\" \u003e(\u003ca  href=\"#\"\u003e3\u003c/a\u003e):         \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                               \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e         \u003cp class=\"noindent\" \u003eThis simplifies to \u003c/p\u003e\u003ctable class=\"equation\"\u003e\u003ctr\u003e\u003ctd\u003e         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                            \u003cmstyle     id=\"x1-42075r6\"  class=\"label\" \u003e\u003c/mstyle\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003c/math\u003e\u003c/td\u003e\u003ctd class=\"eq-no\"\u003e(6)\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eSolve the set of equations with the two new equations. Substitute (\u003ca  href=\"#\"\u003e5\u003c/a\u003e) into         (\u003ca  href=\"#\"\u003e6\u003c/a\u003e):         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                   \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e                                   \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e                                   \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e                                   \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e                                   \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003c/mtd\u003e                                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e         \u003cp class=\"noindent\" \u003eSubstitute this back into (\u003ca  href=\"#\"\u003e5\u003c/a\u003e) and you get: \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                          \u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     4.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eSubstitute the two answers into (\u003ca  href=\"#\"\u003e4\u003c/a\u003e) and solve:         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                    \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-label\"\u003e                            \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-label\"\u003e                                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                   \u003cmtd  class=\"align-label\"\u003e                                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e         \u003cp class=\"noindent\" \u003eThe final answer is then \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                          \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmi  \u003ey\u003c/mi\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmi  \u003ez\u003c/mi\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e         \u003c/dd\u003e\u003c/dl\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e                                                                                                                            \u003cp class=\"indent\" \u003e     "},"image":null,"isRootPath":false,"key":"content/en/encyclopedia/02 Algebra/02 Equations and Inequalities/01 Equations/04 Systems of Equations/407SystemEqsMultipleUnknowns.tex","language":"en","locale":"en","metaDescription":"Learn how to solve systems of three equations with three variables by using the insertion method. This method takes care of the extra variables.","otherLocales":[],"path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/solve-systems-of-equations-with-multiple-unknowns","previewUrl":"https://images.houseofmath.com/eyJrZXkiOiJvdmVybGVhZi1hc3NldHMvcHJldmlldy80MDdzeXN0ZW1lcXNtdWx0aXBsZXVua25vd25zLnBuZyIsImJ1Y2tldCI6ImhvbS1mcm9udGVuZC1hc3NldHMiLCJ2ZXJzaW9uIjozfQ==","sharedLanguageKey":null,"siblings":[{"h1":"What Are Systems of Equations?","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/what-are-systems-of-equations"},{"h1":"Graphic Representation of Equations","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/graphic-representation-of-equations"},{"h1":"How to Graphically Solve Systems of Equations","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/how-to-graphically-solve-systems-of-equations"},{"h1":"Solve Systems of Equations with the Substitution Method","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/solve-systems-of-equations-with-the-substitution-method"},{"h1":"Solve Systems of Equations with the Elimination Method","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/solve-systems-of-equations-with-the-elimination-method"},{"h1":"How to Solve Systems of Nonlinear Equations","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/how-to-solve-systems-of-nonlinear-equations"},{"h1":"Solve Systems of Equations with Multiple Unknowns","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/solve-systems-of-equations-with-multiple-unknowns"}],"taskReference":null,"title":"Solve Systems of Equations with Multiple Unknowns","type":"encyclopedia"},"topicOverview":[{"h1":"Numbers and 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Method","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/solve-systems-of-equations-with-the-elimination-method"},{"h1":"How to Solve Systems of Nonlinear Equations","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/how-to-solve-systems-of-nonlinear-equations"},{"h1":"Solve Systems of Equations with Multiple Unknowns","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/solve-systems-of-equations-with-multiple-unknowns"}]},{"h1":"Rational Equations","path":"/encyclopedia/algebra/equations-and-inequalities/equations/rational-equations"},{"h1":"Radical Equations","path":"/encyclopedia/algebra/equations-and-inequalities/equations/radical-equations"},{"h1":"Equations with Powers","path":"/encyclopedia/algebra/equations-and-inequalities/equations/equations-with-powers"},{"h1":"Techniques","path":"/encyclopedia/algebra/equations-and-inequalities/equations/techniques"}]},{"h1":"Inequalities","path":"/encyclopedia/algebra/equations-and-inequalities/inequalities","shouldRenderChildren":true}]},{"h1":"Logarithms","path":"/encyclopedia/algebra/logarithms","shouldRenderChildren":true}]},{"h1":"Geometry","path":"/encyclopedia/geometry","shouldRenderChildren":true},{"h1":"Statistics and Probability","path":"/encyclopedia/statistics-and-probability","shouldRenderChildren":true},{"h1":"Functions","path":"/encyclopedia/functions","shouldRenderChildren":true},{"h1":"Proofs","path":"/encyclopedia/proofs"}],"previewTopics":[],"_nextI18Next":{"initialI18nStore":{"en":{"imageAlt":{"icons":{"houseOfMath":"Colorful House of Math logo","homShortLogo3D":"White H House of Math logo on multicolored square","chat":"Purple speech balloon chat icon","dashboard":"Purple dashboard 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