How to Graphically Solve Systems of Equations

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Here you will learn how to use graphic solving to solve systems of equations through a set of instructions that will always get you to the finish line!

Rule

Solving a System of Equations Graphically

1.

Solve both equations for

y

. That means you need

y

on its own on the left side in both expressions. It should look like this:

y = a x + b .

2.

Your expressions are now linear functions. Draw them in the same coordinate system.

3.

Find where the two graphs intersect, taking note of one number from the

x

axis and one from the

y

axis—the x- and y- coordinates.

Example 1

Solve the system of equations:

\begin{align} y - 2 x = 2 (1) 4 y + 4 x & = 20 & (2)1. First solve (1) with respect to y : y = 2 x + 2 . Then you solve (2) with respect to y : \begin{array}{l} 4 y + 4 x & = 20 \\ 4 y & = - 4 x + 20 | \div 4 \\ y & = - x + 5 \end{array} 2. Draw the graphs of the expressions for y from the previous point: 3. Find the intersection at x = 1 and y = 4,         and write the answer like this: Answer: (x, y) = (1, 4)Recall that there are three different kinds of solutions. There are solutions where the graphs do not intersect, solutions where the graphs intersect at one point, and solutions where the graphs are on top of each other and intersect at every point.If you’re ever unsure of the solution you find through algebra, you can always get some perspective by drawing the graphs of the equations. I always feel like it helps to have an image of how everything connects together.With training you will find that you can sketch these in your mind, which is very useful!Want to know more? Sign Up It's free!Previous entry Graphic Representation of Equations Next entry Solve Systems of Equations with the Substitution MethodGo backHome All Math Games Statistics ChatAbout 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
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            Norway FAQ Links WeTakeAction Math Magic for Ukraine Tutoring Video Crash Courses Junior Math Math Essentials Encyclopedia Language Norsk Українська Terms Privacy{"props":{"pageProps":{"campaign":false,"topic":{"type":"encyclopedia","language":"en","htmlContent":{"html":" \u003c/p\u003e      \u003cdiv class=\"tcolorbox foobootcamp\" id=\"tcolobox-606\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Video Crash Courses\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003eWant to watch animated videos and solve interactive exercises about solving systems of equations graphically? \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003ca href=\"/bootcamp/algebra/equations/4/2/how\"\u003eClick here to try the Video Crash Course called “Systems of Equations”!\u003c/a\u003e \u003c/p\u003e                                                                                                                     \u003c/div\u003e  \u003c/div\u003e                                                                                                                      \u003cp class=\"indent\" \u003e     Here you will learn how to use \u003ca href=\"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/graphic-representation-of-equations\"\u003e\u003cspan  class=\"ec-lmri-12x-x-144\"\u003egraphic solving\u003c/span\u003e\u003c/a\u003e to solve \u003ca href=\"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/what-are-systems-of-equations\"\u003esystems of equations\u003c/a\u003e through a set of instructions that will always get you to the finish line! \u003c/p\u003e      \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-607\"\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\"\u003eSolving\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ea\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eSystem\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eof\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eEquations\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eGraphically\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e      \u003c/p\u003e\u003cdl class=\"enumerate-enumitem\"\u003e\u003cdt class=\"enumerate-enumitem\"\u003e    1.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eSolve both equations for \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math\u003e.         That means you need \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math\u003e       on its own on the left side in both expressions. It should look like this: \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 \u003cmi  \u003ea\u003c/mi\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\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    2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Your         expressions         are         now         linear         functions.         Draw         them         in         the         same         coordinate         system.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Find         where         the         two         graphs         intersect,         taking         note         of         one         number         from         the         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math\u003e       axis         and         one         from         the         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math\u003e       axis—the         x-         and         y-         \u003cspan  class=\"ec-lmri-12x-x-144\"\u003ecoordinates.\u003c/span\u003e        \u003c/dd\u003e\u003c/dl\u003e                                                                                                       \u003c/div\u003e  \u003c/div\u003e                                                                                                                           \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-608\"\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 system 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\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\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=\"x1-42007r1\"  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\u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e0\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=\"x1-42008r2\"  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\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e\u003c/p\u003e                                                                                                                     \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                           \u003cp class=\"noindent\" \u003e         \u003c/p\u003e\u003cdl class=\"enumerate-enumitem\"\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     1.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eFirst solve (\u003ca  href=\"#\"\u003e1\u003c/a\u003e) with respect to \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math type=\"divide\"\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\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\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         \u003cp class=\"noindent\" \u003eThen you solve (\u003ca  href=\"#\"\u003e2\u003c/a\u003e) with respect to \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math type=\"divide\"\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\u003e4\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\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\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\u003e4\u003c/mn\u003e\u003cmi  \u003ey\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\u003e4\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003cmo  class=\"MathClass-rel\"\u003e|\u003c/mo\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\u003cmi  \u003ey\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\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\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         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eDraw the graphs of the expressions for         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math\u003e         from the previous point:         \u003cdiv class=\"center\"  \u003e \u003cp class=\"noindent\" \u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/2likn2ukjente.svg\" alt=\"Two lines meeting at (1,4)\" width=\"451\" height=\"451\"\u003e\u003c/p\u003e\u003c/div\u003e         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eFind the intersection at \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003c/math\u003e         and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/math\u003e,         and write the answer like this: \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                                            \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003eAnswer:\u003c/mtext\u003e\u003c/mstyle\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-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 \u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\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     Recall that there are three different kinds of solutions. There are solutions where the graphs do not intersect, solutions where the graphs intersect at one point, and solutions where the graphs are on top of each other and intersect at every point. \u003c/p\u003e\u003cp class=\"indent\" \u003e     If you’re ever unsure of the solution you find through algebra, you can always get some perspective by drawing the graphs of the equations. I always feel like it helps to have an image of how everything connects together. \u003c/p\u003e\u003cp class=\"indent\" \u003e     With training you will find that you can sketch these in your mind, which is very useful! \u003c/p\u003e\u003cp class=\"indent\" \u003e     "},"breadcrumbs":[{"h1":"Encyclopedia","_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c","path":"/encyclopedia"},{"path":"/encyclopedia/algebra","h1":"Algebra","_key":"e9327e4b-42fe-571b-bcb9-83fdb63d5f21"},{"path":"/encyclopedia/algebra/equations-and-inequalities","h1":"Equations and Inequalities","_key":"100a0f18-3f76-5653-91ef-a410836c9583"},{"_key":"f99cf246-2b53-56b2-97b1-efb74d01f11e","path":"/encyclopedia/algebra/equations-and-inequalities/equations","h1":"Equations"},{"path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations","h1":"Systems of Equations","_key":"41927a7e-eb51-55a1-908a-da0aea694d01"},{"h1":"How to Graphically Solve Systems of Equations","_key":"72a974e7-9f1f-523b-a5bc-f92de0478676","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/how-to-graphically-solve-systems-of-equations"}],"taskReference":null,"image":{"value":0.75,"url":"https://images.houseofmath.com/overleaf-assets/images/2likn2ukjente.svg","alt":"Two lines meeting at (1,4)","_key":"07340298-44eb-53d8-ab1b-adfdf07a7d98","type":"textwidth","dimension":"width"},"otherLocales":[],"key":"content/en/encyclopedia/02 Algebra/02 Equations and Inequalities/01 Equations/04 Systems of Equations/403MoreGraphicSolvingSysOfEqs.tex","title":"How to Graphically Solve Systems of Equations","previewUrl":"https://images.houseofmath.com/eyJrZXkiOiJvdmVybGVhZi1hc3NldHMvcHJldmlldy80MDNtb3JlZ3JhcGhpY3NvbHZpbmdzeXNvZmVxcy5wbmciLCJidWNrZXQiOiJob20tZnJvbnRlbmQtYXNzZXRzIiwidmVyc2lvbiI6M30=","path":"/encyclopedia/algebra/equations-and-inequalities/equations/systems-of-equations/how-to-graphically-solve-systems-of-equations","h1":"How to Graphically Solve Systems of Equations","locale":"en","metaDescription":"Learn how to solve a system of equations graphically. 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