GM Lesson 058 Solving Linear Equations

Learning Intentions

By the end of this lesson, students will be able to:

• Solve one-step and two-step linear equations.
• Solve linear equations with rational solutions.
• Check solutions by substituting into the original equation.

Prerequisites

Students should already be able to:

• Use inverse operations for addition, subtraction, multiplication and division.
• Substitute numerical values into algebraic expressions.
• Simplify arithmetic involving integers, fractions and decimals.
• Understand that an equation is a statement that two expressions are equal.

Key Idea Summary

A linear equation contains a pronumeral with power , such as , or .

To solve a linear equation, use inverse operations to isolate the unknown. Whatever operation is applied to one side of the equation must also be applied to the other side.

For example, if

then subtracting from both sides gives

A solution can be checked by substituting it back into the original equation.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction

To solve an equation, the goal is to make the unknown the subject.

For a one-step equation, only one inverse operation is needed.

For a two-step equation, usually reverse the operations in this order:

1. Undo addition or subtraction.
2. Undo multiplication or division.

For example, in

the expression means:

• multiply by ,
• then add .

To solve, reverse those operations:

• subtract ,
• then divide by .

Worked Example 1: One-Step Equation with Addition

Solve:

Check:

Therefore, .

Worked Example 2: One-Step Equation with Multiplication

Solve:

Check:

Therefore, .

Worked Example 3: Two-Step Equation

Solve:

Check:

Therefore, .

Worked Example 4: Rational Solution

Solve:

This example has an integer solution.

Now solve:

Check:

Therefore, .

Worked Example 5: Equation Involving Decimals

Solve:

Check:

Therefore, .

Understanding Checks

Check 1

Solve:

Expected answer:

Check 2

Solve:

Expected answer:

Check 3

Solve:

Expected answer:

Check 4

A student solves the equation

and writes:

Identify the error.

Expected response:

The student should subtract from both sides, not add . The correct working is:

Check 5

Check whether is a solution to:

Expected response:

Therefore, is a solution.

Exercises

Simple Familiar Exercises

Exercise 1

Solve:

Exercise 2

Solve:

Exercise 3

Solve:

Exercise 4

Solve:

Exercise 5

Solve:

Exercise 6

Solve:

Exercise 7

Solve:

Exercise 8

Solve:

Complex Familiar Exercises

Exercise 9

Solve:

Exercise 10

Solve:

Exercise 11

Solve:

Exercise 12

Solve:

Exercise 13

Solve:

Exercise 14

Solve:

Exercise 15

Solve and check your solution:

Exercise 16

Solve and check your solution:

Homework Problems

Problem 1

Solve:

Problem 2

Solve:

Problem 3

Solve:

Problem 4

Solve:

Problem 5

Solve:

Problem 6

Solve:

Problem 7

Solve:

Problem 8

Solve:

Problem 9

Solve and check your solution:

Problem 10

Solve and check your solution:

Next: GM Lesson 059 Variables on Both Sides