>Equations>Systems of Equations (Graphing)

Systems of Equations (Graphing)

A system of equations is simply a collection of equations that share a relationship. Here, you’ll learn how to solve two equations with two unknowns—two variables. You’ll learn three methods to solve systems of equations: Graphing, the substitution method, and the elimination method. Let’s start with graphing.

Graphing

Rule

Graphing

1.: Solve both equations for $y$ . This means that you should have $y$ by itself on one side of both expressions. It looks something like this: $y = a x + b$ .
2.: Your expressions are now linear functions. Draw both of them in the same coordinate system.
3.: Find the point of intersection and write down the corresponding values from the $x$ -axis and the $y$ -axis.

Example 1

Solve the system of equations

\begin{align} y - 2 x = 2 (1) 4 y + 4 x & = 20 & (2)First you 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} You may now plot the lines y = 2 x + 2 and y = - x + 5 : The point of intersection gives you x = 1 and y = 4 . ANSWER: (x, y) = (1, 4)Want to know more? Sign Up It's free!Previous entry Activities 8 Next entry Systems of Equations (Substitution)Home All Math Games Progress LeagueAbout 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 Bootcamps Junior Math Pen & Paper exercises Encyclopedia Book a Tutor Language Norsk Українська Terms Privacy{"props":{"pageProps":{"campaign":false,"topic":{"locale":"en","language":"en","previewUrl":"https://images.houseofmath.com/eyJrZXkiOiJvdmVybGVhZi1hc3NldHMvcHJldmlldy8xN3N5c3RlbXNvZmVxdWF0aW9ucy5wbmciLCJidWNrZXQiOiJob20tZnJvbnRlbmQtYXNzZXRzIiwidmVyc2lvbiI6M30=","htmlContent":{"html":"                                                                                                                                                                                                                                          \u003c/p\u003e\u003cp class=\"indent\" \u003e     A \u003cspan  class=\"ec-lmri-12x-x-144\"\u003esystem of equations \u003c/span\u003eis simply a collection of equations that share a relationship. Here, you’ll learn how to solve two equations with two unknowns—two variables. You’ll learn three methods to solve systems of equations: \u003cspan  class=\"ec-lmri-12x-x-144\"\u003eGraphing\u003c/span\u003e, the \u003ca href=\"/drill/equations/systems-of-equations-substitution\"\u003esubstitution method\u003c/a\u003e, and the \u003ca href=\"/drill/equations/systems-of-equations-elimination\"\u003eelimination method\u003c/a\u003e. Let’s start with graphing. \u003c/p\u003e\u003cp class=\"noindent\" \u003e \u003c/p\u003e \u003cp class=\"indent\" \u003e     \u003ch2 class=\"section-title\"\u003e \u003ca   id=\"x1-376000\"\u003e\u003c/a\u003eGraphing\u003c/h2\u003e \u003c/p\u003e      \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-175\"\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\"\u003eGraphing\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\"\u003e         Solve         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.         This         means         that         you         should         have         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math\u003e       by         itself         on         one         side         of         both         expressions.         It         looks         something         like         this:         \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 \u003cmi  \u003ea\u003c/mi\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/math type=\"punct\"\u003e.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Your         expressions         are         now         \u003ca href=\"/drill/functions/linear-functions\"\u003elinear         functions\u003c/a\u003e.         Draw         both         of         them         in         the         same         coordinate         system.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Find         the         point         of         intersection         and         write         down         the         corresponding         values         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         the         \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/math\u003e-axis.         \u003c/dd\u003e\u003c/dl\u003e                                                                                                                      \u003c/div\u003e  \u003c/div\u003e                                                                                                                           \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-176\"\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=\"x1-376006r1\"  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=\"x1-376007r2\"  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\" \u003eFirst you 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: \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\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\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 \u003cp class=\"noindent\" \u003eYou may now plot the lines \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\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e2\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 \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\u003c/math type=\"divide\"\u003e: \u003c/p\u003e \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 linear graphs intersecting at (1,4)\" width=\"451\" height=\"451\"\u003e\u003c/p\u003e\u003c/div\u003e \u003cp class=\"noindent\" \u003eThe point of intersection gives you \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 type=\"punct\"\u003e. \u003c/p\u003e\u003cp class=\"noindent\" \u003eANSWER: \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \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/p\u003e                                                                                                                     \u003c/div\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                                      \u003cp class=\"indent\" \u003e     "},"siblings":[{"h1":"What Is an Equation?","path":"/drill/equations/what-is-an-equation"},{"h1":"Activities 1","path":"/drill/equations/equations-activities-1"},{"h1":"Remove Number in Front of x","path":"/drill/equations/remove-number-in-front-of-x"},{"h1":"Activities 2","path":"/drill/equations/equations-activities-2"},{"h1":"Remove Number Under x","path":"/drill/equations/remove-number-under-x"},{"h1":"Activities 3","path":"/drill/equations/equations-activities-3"},{"h1":"Equations with Several Terms on Both Sides","path":"/drill/equations/equations-with-several-terms-on-both-sides"},{"h1":"Activities 4","path":"/drill/equations/equations-activities-4"},{"h1":"Equations with Fractions and Parentheses","path":"/drill/equations/equations-with-fractions-and-parentheses"},{"h1":"Activities 5","path":"/drill/equations/equations-activities-5"},{"h1":"Equations with x in the Denominator","path":"/drill/equations/equations-with-x-in-the-denominator"},{"h1":"Activities 6","path":"/drill/equations/equations-activities-6"},{"h1":"Checking the Answer","path":"/drill/equations/checking-the-answer"},{"h1":"Activities 7","path":"/drill/equations/equations-activities-7"},{"h1":"How to Solve Power Equations","path":"/drill/equations/how-to-solve-power-equations"},{"h1":"Activities 8","path":"/drill/equations/equations-activities-8"},{"h1":"Systems of Equations (Graphing)","path":"/drill/equations/systems-of-equations-graphing"},{"h1":"Systems of Equations 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