154. Solving Equations by Expanding Brackets

Learning Intentions

• To understand that the distributive law can be used to Expand brackets within equations
• Solve equations by expanding brackets

Pre-requisite Summary

• The distributive law allows a factor outside brackets to multiply every term inside the brackets
• Expanding brackets creates an equivalent expression
• An equation is a mathematical statement that two expressions are equal
• Equivalent equations have the same solution
• Like terms can be combined after expanding
• The same term can be added to or subtracted from both sides of an equation to keep the equation balanced
• The same non-zero number can multiply or divide both sides of an equation to produce an equivalent equation
• A solution should 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 .

Problems

Problem 1

Solve .

Problem 2

Solve .

Problem 3

Solve .

Problem 4

Solve .

Problem 5

Solve .

Problem 6

Solve .

Exercises

Understanding and Fluency

Exercise 1.

Expand each side of the equation.

a)

b)

c)

Exercise 2.

Expand each side of the equation.

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 each equation.

a)

b)

c)

Exercise 8.

Solve each equation.

a)

b)

c)

Reasoning

Exercise 9.

Explain why expanding both sides of an equation produces an equivalent equation.

Exercise 10.

A student writes as . Explain the error.

Exercise 11.

Noah says the first step in solving is to divide both sides by . Explain why expanding first is more helpful.

Exercise 12.

Explain why solving requires both expanding and then solving an equation with pronumerals on both sides.

Problem-solving

Exercise 13.

A gym charges $2 per class plus a $3 joining fee. One person pays $17 in total. Write and solve an equation to find the number of classes attended.

Exercise 14.

A rectangle has width cm and fixed length cm. Its area is . Write and solve an equation to find .

Exercise 15.

A builder makes identical shelves, each of length metres. The total length of timber used is metres. Write and solve an equation to find .

Exercise 16.

A school hires buses. Each bus holds students. If buses carry students altogether, write and solve an equation to find .

Potential Misunderstandings

• Thinking the distributive law only applies in expressions and not in equations
• Multiplying the outside factor by only the first term inside the brackets
• Forgetting to multiply both the number and the pronumeral terms inside the brackets
• Writing as instead of
• Expanding correctly but then making errors when combining like terms
• Thinking that expanding changes the solution of the equation rather than creating an equivalent equation
• Trying to solve the equation before removing the brackets when expansion would simplify the structure
• Making sign errors when expanding brackets with subtraction, such as
• Forgetting that after expanding, the equation may still need terms collected on one side
• Failing to check the solution in the original equation

Next: 155e. Solving Simple Quadratic Equations