# 7.1: Linear Systems by Graphing

**At Grade**Created by: CK-12

## 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.

## Vocabulary

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.

Example:

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

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

Solution:

The problem can be translated as:

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

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*}

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*}

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

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

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

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

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

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

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

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

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

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

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*} values, as for a table, have students solve each equation for \begin{align*}y\end{align*} 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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