153. Solving Equations with Pronumerals on Both Sides

Learning Intentions

• To understand that the same term can be subtracted from or added to both sides of an equation to produce an equivalent equation
• Solve equations involving pronumerals on both sides

Pre-requisite Summary

• An equation is a statement that two expressions are equal
• Equivalent equations have the same solution
• Adding the same term to both sides of an equation preserves equality
• Subtracting the same term from both sides of an equation preserves equality
• Like terms can be combined to Simplify expressions
• Solving an equation means isolating the pronumeral
• Solutions can be checked by substitution into the original equation

Worked Examples

Worked Example 1

Solve .

Worked Example 2

Solve .

Worked Example 3

Solve .

Worked Example 4

Solve .

Worked Example 5

Solve .

Worked Example 6

Solve .

Worked Example 7

Solve .

Worked Example 8

Solve .

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 .

Exercises

Understanding and Fluency

Exercise 1.

Write an equivalent equation by subtracting the same term from both sides.

a)

b)

c)

Exercise 2.

Write an equivalent equation by adding the same term to both sides.

a)

b)

c)

Exercise 3.

Solve each equation.

a)

b)

c)

Exercise 4.

Solve each equation.

a)

b)

c)

Exercise 5.

Solve each equation.

a)

b)

c)

Exercise 6.

Solve each equation.

a)

b)

c)

Exercise 7.

Solve and Check each equation.

a)

b)

c)

Exercise 8.

Solve and check each equation.

a)

b)

c)

Reasoning

Exercise 9.

Explain why subtracting from both sides of gives an equivalent equation.

Exercise 10.

A student says that in , the terms can be cancelled immediately to give . Explain the error.

Exercise 11.

Explain why adding or subtracting the same term from both sides does not change the solution of an equation.

Exercise 12.

Noah solves by subtracting from the left side only. Explain why this does not produce an equivalent equation.

Problem-solving

Exercise 13.

Two mobile plans cost the same amount when compared for a certain number of months. One plan costs dollars and the other costs dollars. Write and solve an equation to Solve .

Exercise 14.

A student has two expressions for the same perimeter: and . Write and solve an equation to find .

Exercise 15.

A shop has two discount rules that give the same final price: dollars and dollars. Write and solve an equation to find .

Exercise 16.

A builder measures the same length in two ways: cm and cm. Write and solve an equation to find .

Potential Misunderstandings

• Thinking a term can be added or subtracted from one side only and still keep the equation balanced
• Believing that moving a term across the equals sign is a separate rule rather than adding or subtracting the same term to both sides
• Forgetting that the whole term must be added or subtracted, including its sign
• Combining unlike terms when simplifying each side
• Cancelling terms incorrectly across the equals sign
• Making sign errors when subtracting a pronumeral term from both sides
• Solving part of the equation correctly but not continuing until the pronumeral is isolated
• Forgetting to check a solution by substitution into the original equation
• Thinking that equations with pronumerals on both sides need a different method from ordinary equations, rather than the same idea of equivalent equations

Next: 154. Solving Equations by Expanding Brackets