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You are reading an older version of this FlexBook® textbook: CK-12 Algebra I Teacher's Edition Go to the latest version.

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

Additional Example:

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.”

Additional Examples:

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.

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