Use inverse operations to solve equations, keep both sides balanced, and check solutions by substitution.
Today's hook, A cinema ticket booking costs $\$5$ plus $\$12$ per ticket. If the total is $41, how many tickets were purchased? Can you figure it out without trial and error?
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Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
A cinema ticket booking costs $\$5$ plus $\$12$ per ticket. If the total is $41, how could you find the number of tickets? Without algebra write your first strategy.
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Inverse operations, the key to solving
+5 XP to read An equation is like a balance: both sides have the same value.
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Inverse operations, the key to solving
+5 XP to read
An equation is like a balance: both sides have the same value. To find the unknown, you undo the operations that were done to the variable, this is called using inverse operations.
One-step equations: $x + a = b$, subtract $a$ from both sides. $ax = b$, divide both sides by $a$.
Two-step equations: $ax + b = c$, subtract $b$ first, then divide by $a$.
Whatever you do to one side, you must do to the other
Inverse operations
Addition undone by subtraction. Multiplication undone by division. Apply the inverse to both sides.
Order for two-step
In $3x + 5 = 26$, undo addition first ($-5$), then undo multiplication ($\div 3$). Work backwards through the order of operations.
Always check
Substitute your solution back into the original equation. If both sides are equal, the solution is correct.
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What you'll master
Know
Key facts
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
Concepts
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
Skills
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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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 $3x + 5$.
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Equations must stay balanced
An equation is like a balance: both sides have the same value.
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Equations must stay balanced
core concept
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.
Equation = balance: any operation applied to one side must be applied to the other. To solve, apply the inverse operations in reverse order. One-step equations: one inverse operation. Always check by substituting the solution back.
Pause, copy the balance principle (any operation applied to one side must be applied to the other) and the inverse-operation rule (to isolate a variable, apply the inverse of each operation currently attached to it) into your book.
Did you get this? True or false: to solve $x + 9 = 15$, you should add 9 to both sides.
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Two-step equations, undo the outside operation first
We just saw that solving an equation means keeping both sides balanced, apply the same inverse operation to both sides to...
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Two-step equations, undo the outside operation first
core concept
We just saw that solving an equation means keeping both sides balanced, apply the same inverse operation to both sides to isolate the variable. That raises a question: what if the variable has two operations applied to it, like 3x + 5 = 26, which operation do you undo first? This card answers it → undo operations in reverse BODMAS order: undo addition/subtraction first, then multiplication/division, for 3x + 5 = 26, subtract 5 first, then divide by 3.
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
Solves the equation
$x \rightarrow 3x \rightarrow 3x + 5$
$3x + 5 \rightarrow 3x \rightarrow x$
Common error. Do not divide by 3 first in $3x + 5 = 26$. The $+5$ is attached after the multiplication, so it is undone first.
Key inverse operations reference:
Equation type
Step to isolate variable
$x + a = b$
subtract $a$ from both sides
$ax = b$
divide both sides by $a$
$ax + b = c$
subtract $b$, then divide by $a$
Two-step equations: undo the operations in reverse order of BODMAS. For 3x + 5 = 26: subtract 5 first (undo addition), then divide by 3 (undo multiplication). Always undo the operation furthest from the variable last.
Pause, copy the two-step solve order: undo addition/subtraction first, then undo multiplication/division, e.g. 3x + 5 = 26: subtract 5 → divide by 3 → x = 7 into your book.
Quick check: To solve $4x + 7 = 31$, what is the correct first step?
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Equations with brackets
We just saw two-step equations using reverse BODMAS, subtract or add first, then multiply or divide.
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Equations with brackets
core concept
We just saw two-step equations using reverse BODMAS, subtract or add first, then multiply or divide. That raises a question: what changes when the variable is inside a bracket with a coefficient outside, like 3(2x + 1) = 21? This card answers it → either divide both sides by the coefficient first (if it divides evenly), or expand the bracket and then solve, both methods give the same answer.
For an equation like $2(x + 4) = 18$, you can divide both sides by the coefficient of the bracket first, or expand the bracket first.
Method A, divide first
Method B, expand first
$2(x + 4) = 18$ $x + 4 = 9$ $x = 5$
$2x + 8 = 18$ $2x = 10$ $x = 5$
Both methods give the same answer. Choose whichever is easier for the numbers in the question.
For equations with brackets, either divide both sides by the coefficient first or expand the bracket first. Both methods give the same answer. Choose: divide first if the coefficient divides evenly; expand first otherwise.
Pause, copy both bracket strategies: (1) divide out the coefficient first if it divides evenly; (2) expand first then solve, and note when each is more efficient into your book.
Fill the gap: To solve $3x + 5 = 26$: first subtract from both sides to get $3x =$ , then divide by 3 to get $x =$ .
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Worked examples · 3 in a row, reveal as you go
Worked examples · 3 in a row, reveal as you go
Worked examples · 3 in a row, reveal as you go
PROBLEM 1 · ONE-STEP EQUATIONS
Solve: (a) $x + 7 = 19$ (b) $4x = 36$
a
$x + 7 - 7 = 19 - 7$
Subtract 7 from both sides (inverse of adding 7).
a
$x = 12$
Check: $12 + 7 = 19$ ✓
b
$\dfrac{4x}{4} = \dfrac{36}{4}$, so $x = 9$
Divide both sides by 4 (inverse of multiplying by 4). Check: $4 \times 9 = 36$ ✓
PROBLEM 2 · SOLVE AND CHECK $3x + 5 = 26$
Solve $3x + 5 = 26$ and verify the solution by substitution.
1
$3x + 5 - 5 = 26 - 5$
Subtract 5 from both sides first (undo the addition).
2
$3x = 21$, so $x = 7$
Divide both sides by 3 (undo the multiplication).
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Check: $3(7) + 5 = 21 + 5 = 26$ ✓
Substitute $x = 7$ back into the original equation. Both sides equal 26, so the solution is correct.
PROBLEM 3 · EQUATION WITH BRACKETS
Solve $2(x + 4) = 18$.
1
$2(x + 4) = 18$, so $x + 4 = 9$
Divide both sides by 2 (the coefficient of the bracket).
2
$x = 9 - 4 = 5$
Subtract 4 from both sides.
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Check: $2(5 + 4) = 2(9) = 18$ ✓
Alternative: expand first, $2x + 8 = 18 \Rightarrow 2x = 10 \Rightarrow x = 5$. Either method is valid.
Odd one out: Three of these are correct statements about solving equations. Which one is wrong?
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Common errors · the 3 traps that cost marks
Common errors · the 3 traps that cost marks
Common errors · the 3 traps that cost marks
Trap 01
Dividing before undoing addition
In $3x + 5 = 26$, dividing by 3 first gives $x + \frac{5}{3} = \frac{26}{3}$, which is messy and error-prone. The $+5$ is the outside operation, undo it first to keep the numbers clean.
Trap 02
Applying the operation to only one side
Writing $3x + 5 - 5 = 26$ (without subtracting 5 from the right side too) breaks the balance. Every balancing step must be applied equally to both sides of the equation.
Trap 03
Skipping the substitution check
Many marks are lost by not checking. Substituting $x = 7$ back into $3(7) + 5 = 26$ takes five seconds and can catch a sign error or arithmetic mistake that would otherwise cost full marks.
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Revisit the ticket problem
Quick-fire practice · 4 equations
Quick-fire practice · 4 equations
1
Solve $y - 6 = 15$ and check your answer.
2
Solve $5a = 45$ and check your answer.
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Solve $2m + 7 = 31$ and check your answer.
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Solve $4(p - 3) = 28$ and check your answer.
Teach it back: In one or two sentences, explain why you undo addition before division when solving a two-step equation like $3x + 5 = 26$.
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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: $t = 3$.
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Show what you have learned
Multiple choice, then short answer under exam conditions.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
ApplyBand 32 marks
Q1. Solve $6x = 54$ and check your solution. (2 marks)
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ApplyBand 43 marks
Q2. Solve $4x + 9 = 37$ and check your solution by substitution. (3 marks)
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AnalyseBand 44 marks
Q3. 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)
