Materials

  • CCB Mathematics pages 184 - 189

Standards

  • Analyze and solve linear equations and pairs of simultaneous linear equations.

Objectives

  • Solve systems of two linear equations

  • Interpret graphs of two linear equations

  • Use linear equations to solve problems

Key Concept

  • A pair of linear equations forms a system of two simultaneous linear equations. The solution to a system of two linear equations in two variables corresponds to a point of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Vocabulary

Tier 2 Simultaneous
Tier 3 Addition Method
Substitution Method
System of Linear equations
Test Words Eliminate

Before Lesson

In this lesson, students learn to graph two simultaneous linear equations. To determine student readiness, distribute graph paper and have students draw a coordinate plane. Give them two points on the plane and have students to use the points to determine and graph an equation of the line. Observe students as they work to determine if more practice would be helpful before beginning the lesson.

Background

Tell students that two variables have a linear relationship if their corresponding points lie on the same line in a coordinate plane. Every pair of coordinate points on a line is a solution to a linear equation that represents the line. Have students graph a line, for example, y=3x-1. Then have students find a point that is on the line, (2, 5) for example, and have them substitute the value of x and y into the equation to see that the point is a solution to that equation.

Guided Practice

  • Simultaneous Linear Equations
  • Combining Methods to Solve Pairs of Linear Equations

Core Skill

Solve Pairs of Linear Equations: Complete the exercise in applying the substitution method for solving a pair of equations without graphing as a class, or have students work through the process independently or in small groups as you observe. Have students apply the substitution method to the equations at the bottom of the sidebar. Then ask them to explain how they applied the method to find the solution to the pair.

Solve Simple Equations by Inspection: Read the text with students to help them recognize two special cases in which a single solution for a pair of simultaneous equations cannot be obtained. Emphasize that inspection alone is enough to reach this conclusion. In the first example, the left-hand side of both equations is the same (3p+2q), but the right-hand side is different. Plotting these equations would result in parallel lines. Because the lines do not intersect, there is no solution. In the second example, the first equation is a multiple of the second equation, thus making the equations equivalent. This results in identical equations, which have infinite solutions. Guide students through a discussion of the third case.

Extension

Identify Multiple Meanings: Write the word equilibrium on the board. Ask students to use a print or online dictionary to define the word. Then explain the word’s meaning as it applies to children on opposite sides of a seesaw or a circus performer on a high wire. Also explain how equilibrium applies to one’s mental health or between two sides of an argument. Finally, ask students to explain the word’s meaning as it applies to mathematics.

Use Concepts to Solve Non-Routine Problems: Have students work collaboratively to generate examples of when the intersection of two graphs has practical relevance, such as the point at which the orbit of the earth and the orbit of a meteorite intersect, indicating a point at which the meteorite falls to Earth. Encourage students to research their topic and graph lines to indicate an equilibrium point.