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# 7.1: Linear Systems by Graphing

Difficulty Level: 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, x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}, such that their sum is 10\begin{align*}10\end{align*} and their difference is 4\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:

x+yxy=10=4

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

x567y543sum101010difference024

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

2x+yxy=8=1

x0123y

x0123y

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

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

a.

x+y2x+y=3=1

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

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

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

b.

7x+y3x2y=7=14

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

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

iii. (4,3)\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 y\begin{align*}y\end{align*} values, as for a table, have students solve each equation for y\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.

## Date Created:

Feb 22, 2012

Aug 22, 2014
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