# 7.1: Linear Systems by Graphing

## 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,* *and* , *such that their sum is* *and their difference is* .

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:

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

Additional Example:

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

Hint: Solve each equation for 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*and 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.

i.

ii.

iii.

b.

i.

ii.

iii.

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