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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, $x$ and $y$, such that their sum is $10$ and their difference is $4$.

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

Solution:

The problem can be translated as:

$x+y & = 10 \\x-y & = 4$

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

$x && y && \text{sum} && \text{difference}\\5 && 5 && 10 && 0 \\6 && 4 && 10 && 2 \\7 && 3 && 10 && 4 \surd$

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

$2x +y & = 8 \\x - y & = 1$

$x && y \\0 && \\1 && \\2 && \\3 &&$

$x && y \\0 && \\1 && \\2 && \\3 &&$

Hint: Solve each equation for $y$ 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 $x$ and $y$ 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.”

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

a. $x+y & =3 \\2x + y & = 1$

i. $(2,1)$

ii. $(5,-2)$

iii. $(-2,5) \surd$

b. $7x+y & =7 \\-3x-2y & =-14$

i. $(1,0)$

ii. $(0,7) \surd$

iii. $(4,3)$

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 $y$ values, as for a table, have students solve each equation for $y$ 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.

Feb 22, 2012

Aug 22, 2014

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