Solving One-Step and Two-Step Equations

Use inverse operations to solve equations, keep both sides balanced, and check solutions by substitution.

45 min Algebra Formulas and equations Lesson 2 of 13
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Think First

A cinema ticket booking costs $5 plus $12 per ticket. If the total is $41, how could you find the number of tickets?

Type your first strategy below. You will refine it using inverse operations.

Write your first strategy in your book. You will refine it using inverse operations.

Write your first strategy in your book
Saved

Know

  • An equation states that two sides have equal value.
  • Inverse operations undo each other.
  • A solution is a value that makes the equation true.

Understand

  • Whatever is done to one side of an equation must be done to the other side.
  • Two-step equations are solved by undoing addition or subtraction before multiplication or division.
  • Substitution checks whether the solution is correct.

Can Do

  • Solve one-step equations such as $x + 7 = 19$ and $4x = 36$.
  • Solve two-step equations such as $3x + 5 = 26$.
  • Explain and check each solution.
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Inverse Operations

$x + a = b$
subtract $a$ from both sides
$ax = b$
divide both sides by $a$
$ax + b = c$
subtract $b$, then divide by $a$

Core Content

Key Terms
EquationA statement showing two expressions are equal.
SolutionA value that makes an equation true.
Inverse operationAn operation that undoes another operation, such as subtraction undoing addition.
BalanceThe idea that both sides of an equation must remain equal.
Substitution checkReplacing the variable with the proposed solution to test the equation.
CoefficientThe number multiplying the variable, such as 3 in $3x + 5$.

1. Equations Must Stay Balanced

An equation is like a balance: both sides have the same value.

To keep the equation true, any operation used on one side must also be used on the other side. This is why each solving step changes both sides in the same way.

Key idea: Solving means isolating the variable while keeping the two sides equal.
Worked Example 1

Solve one-step equations

a. Solve $x + 7 = 19$.

$x + 7 - 7 = 19 - 7$

$x = 12$

b. Solve $4x = 36$.

$\frac{4x}{4} = \frac{36}{4}$

$x = 9$

2. Two-Step Equations Undo the Outside Operation First

For $3x + 5 = 26$, the variable is first multiplied by 3, then 5 is added.

When solving, work backwards: undo the addition first, then undo the multiplication.

Builds the expression

$x \rightarrow 3x \rightarrow 3x + 5$

Solves the equation

$3x + 5 \rightarrow 3x \rightarrow x$

Common error: Do not divide by 3 first in $3x + 5 = 26$. The $+5$ is attached after the multiplication, so it is undone first.
Worked Example 2

Solve and check $3x + 5 = 26$

$3x + 5 = 26$

$3x + 5 - 5 = 26 - 5$

$3x = 21$

$x = 7$

Check: $3(7) + 5 = 21 + 5 = 26$, so the solution is correct.

Worked Example 3

Solve an equation with brackets

Solve $2(x + 4) = 18$.

$2(x + 4) = 18$

$x + 4 = 9$

$x = 5$

Check: $2(5 + 4) = 2(9) = 18$.

Alternative: You could expand first: $2x + 8 = 18$, then $2x = 10$, so $x = 5$.
Activity

Equation Practice

  1. Solve $y - 6 = 15$ and check your answer.
  2. Solve $5a = 45$ and check your answer.
  3. Solve $2m + 7 = 31$ and check your answer.
  4. Solve $4(p - 3) = 28$ and check your answer.
Complete the equation practice in your book.

Revisit the Ticket Problem

The cinema problem can be modelled by $5 + 12t = 41$, where $t$ is the number of tickets. Subtract 5 from both sides, then divide by 12: $t = 3$.

Explain the solving order in your book.

Assessment

MC

Multiple Choice

Random questions from the lesson bank - feedback appears immediately.

SA

Short Answer

Show each inverse operation and check at least one solution.

1. Solve $6x = 54$ and check your solution. 2 MARKS

Answer in your book.

2. Solve $4x + 9 = 37$ and check your solution by substitution. 3 MARKS

Answer in your book.

3. A taxi fare is $7 plus $3 per kilometre. The total fare is $31. Write and solve an equation for the number of kilometres. 4 MARKS

Answer in your book.
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