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7.1: Linear Systems by Graphing

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

At the end of this lesson, students will be able to:

  • Determine whether an ordered pair is a solution to a system of equations.
  • Solve a system of equations graphically.
  • Solve a system of equations graphically with a graphing calculator.
  • Solve word problems using systems of equations.


Terms introduced in this lesson:

system of equations
solution to an equation
solution to a system of equations
point of intersection

Teaching Strategies and Tips

Present students with a basic problem to motivate systems of equations.


Find two numbers, \begin{align*}x\end{align*}x and \begin{align*}y\end{align*}y, such that their sum is \begin{align*}10\end{align*}10 and their difference is \begin{align*}4\end{align*}4.

Allow students some time to find the numbers. Encourage guess-and-check at first. A good place to start is with pairs of integers.


The problem can be translated as:

\begin{align*}x+y & = 10 \\ x-y & = 4\end{align*}x+yxy=10=4

Ask: Of all the possible ordered pair solutions to the first equation, which also satisfy the second?

\begin{align*}x && y && \text{sum} && \text{difference}\\ 5 && 5 && 10 && 0 \\ 6 && 4 && 10 && 2 \\ 7 && 3 && 10 && 4 \surd\end{align*}x567y543sum101010difference024

Additional Example:

Complete the table for each equation. Compare the rows of the two tables to determine the solution to the system.

\begin{align*}2x +y & = 8 \\ x - y & = 1\end{align*}2x+yxy=8=1

\begin{align*}x && y \\ 0 && \\ 1 && \\ 2 && \\ 3 && \end{align*}x0123y

\begin{align*}x && y \\ 0 && \\ 1 && \\ 2 && \\ 3 && \end{align*}x0123y

Hint: Solve each equation for \begin{align*}y\end{align*}y first.

Use the introduction and Example 1 to point out that a system of equations is one problem despite there being two equations.

  • The two equations must be solved “together” or simultaneously.
  • The problem is not done until both \begin{align*}x\end{align*}x and \begin{align*}y\end{align*}y have been determined.
  • The solution to an equation is a number; the solution to a system of equations is an ordered pair.
  • An ordered pair solution satisfies, or “makes the equations true.”

Additional Examples:

Find the solution to the following systems of equations by checking each of the choices in the list.

a. \begin{align*}x+y & =3 \\ 2x + y & = 1\end{align*}x+y2x+y=3=1

i. \begin{align*}(2,1)\end{align*}(2,1)

ii. \begin{align*}(5,-2)\end{align*}(5,2)

iii. \begin{align*}(-2,5) \surd \end{align*}(2,5)

b. \begin{align*}7x+y & =7 \\ -3x-2y & =-14\end{align*}7x+y3x2y=7=14

i. \begin{align*}(1,0)\end{align*}(1,0)

ii. \begin{align*}(0,7) \surd\end{align*}(0,7)

iii. \begin{align*}(4,3)\end{align*}(4,3)

Use Examples 2-4 to demonstrate the graphing method for solving a system of equations.

  • Lines can be graphed using any method: constructing a table of values, graphing equations in slope-intercept form, solving for and plotting the intercepts.

Emphasize that the graphing method approximates solutions.

  • It is exact when the point of intersection has integer coordinates or easily discernible rational numbers.
  • Suggest that students draw careful graphs.
  • By zooming in, a calculator provides the coordinates of the intersection point to any degree of accuracy although the solution can still be approximate.

Error Troubleshooting

General Tip: To generate \begin{align*}y\end{align*}y values, as for a table, have students solve each equation for \begin{align*}y\end{align*}y first. See Example 6.

General Tip: To demonstrate that an ordered pair is a solution to a system, remind students that it must satisfy both equations. To demonstrate that an ordered pair is not a solution to a system, remind students that at least one of the equations will not be satisfied.

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