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Lesson 16 ~35 min Unit 2 · Linear Relationships +100 XP

Solving Linear Equations Review

Master the balance method — use inverse operations to solve one-step, two-step, brackets and unknowns-on-both-sides equations.

Today's hook: An equation is like a balanced scale. To find the unknown, use inverse operations — but always do the same thing to both sides.
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Think First
warm-up

Solve $2x + 5 = 17$. What operation do you do first? Why?

First subtract 5 from both sides (undo $+5$): $2x = 12$. Then divide by 2: $x = 6$.
1
One-Step Equations
+5 XP

These require just one inverse operation to solve.

TypeExampleOperationSolution
$x + a = b$$x + 6 = 13$Subtract 6 from both sides$x = 7$
$x - a = b$$x - 4 = 9$Add 4 to both sides$x = 13$
$ax = b$$3x = 21$Divide both sides by 3$x = 7$
$\frac{x}{a} = b$$\frac{x}{5} = 4$Multiply both sides by 5$x = 20$
Example A — Solve $x - 8 = 15$
  1. Add 8 to both sides (inverse of $-8$)

    $$x - 8 + 8 = 15 + 8$$

    $$x = 23$$

  2. Check: substitute $x = 23$ back

    $$23 - 8 = 15 \checkmark$$

$x = 23$
Example B — Solve $\tfrac{x}{4} = 7$
  1. Multiply both sides by 4 (inverse of $\div 4$)

    $$\frac{x}{4} \times 4 = 7 \times 4$$

    $$x = 28$$

  2. Check: $\frac{28}{4} = 7$

    $$\checkmark$$

$x = 28$
$2x + 5$ 17 Balanced! $-5$ $-5$
2
Two-Step Equations
+5 XP

These require two inverse operations. Follow reverse BODMAS:

  1. Undo addition or subtraction first
  2. Then undo multiplication or division
Original: $3x + 4$ → Step 1: Undo $+4$ (subtract 4) Step 2: Undo $\times 3$ (divide by 3) → Solution: $x = 4$ → Check
Worked Example — Solve $3x + 4 = 16$
  1. Subtract 4 from both sides (undo $+4$)

    $$3x + 4 - 4 = 16 - 4$$

    $$3x = 12$$

  2. Divide both sides by 3 (undo $\times 3$)

    $$\frac{3x}{3} = \frac{12}{3}$$

    $$x = 4$$

  3. Check: $3(4) + 4 = 12 + 4 = 16$

    $$\checkmark$$

$x = 4$
3
Equations with Brackets
+5 XP

When an equation contains brackets, expand first, then solve as normal.

$2(x + 3) = 14$ → Expand: $2x + 6 = 14$ → Solve: $x = 4$
Worked Example — Solve $2(x + 3) = 14$
  1. Expand the brackets

    $$2 \times x + 2 \times 3 = 14$$

    $$2x + 6 = 14$$

  2. Subtract 6, then divide by 2

    $$2x = 8 \implies x = 4$$

  3. Check: $2(4 + 3) = 2 \times 7 = 14$

    $$\checkmark$$

Always expand brackets before collecting like terms.
Tip
Always expand brackets first before collecting like terms or isolating the unknown.
4
Unknowns on Both Sides
+5 XP

When the unknown appears on both sides, collect all $x$ terms on one side and all numbers on the other.

Worked Example — Solve $3x + 5 = 2x + 11$
  1. Subtract $2x$ from both sides

    $$3x + 5 - 2x = 2x + 11 - 2x$$

    $$x + 5 = 11$$

  2. Subtract 5 from both sides

    $$x = 6$$

  3. Check

    LHS: $3(6) + 5 = 23$. RHS: $2(6) + 11 = 23$. $\checkmark$

Move the smaller $x$ term to avoid negative coefficients.
5
Checking Solutions
always do this!

Always substitute your answer back into the original equation to verify.

Example — Check if $x = 3$ is the solution to $5x - 7 = 8$
  1. Substitute $x = 3$ into the LHS

    $$\text{LHS} = 5(3) - 7 = 15 - 7 = 8$$

    RHS $= 8$

  2. Compare: LHS $=$ RHS

    $$8 = 8 \quad \checkmark$$

    The solution $x = 3$ is correct.

If LHS ≠ RHS after substitution, you have an error. Go back and find it!
6
Common Equation Types — Summary
reference
Equation TypeExampleStrategyKey Step
One-step$x + 5 = 12$Single inverse operationSubtract 5
Two-step$2x + 3 = 11$Reverse BODMASSubtract 3, then divide by 2
With brackets$3(x - 2) = 15$Expand first, then solveDistribute the 3
Unknowns both sides$4x + 2 = 2x + 10$Collect $x$ on one sideSubtract $2x$
Fractions$\frac{x}{4} + 2 = 5$Undo addition, then multiplySubtract 2, then multiply by 4
Golden Rule
Do the same operation to both sides of the equation.
Reverse BODMAS
Undo addition/subtraction first, then multiplication/division.
Always check
Substitute your answer back into the original equation.
Brain Trainer
speed drill

Set a timer for 3 minutes. Solve as many as you can!

1

$x + 9 = 15$

Answer

$x = 6$

2

$x - 6 = 4$

Answer

$x = 10$

3

$5x = 35$

Answer

$x = 7$

4

$\frac{x}{4} = 6$

Answer

$x = 24$

5

$2x + 5 = 17$

Answer

$x = 6$

6

$3x - 7 = 14$

Answer

$x = 7$

7

$4(x + 2) = 24$

Answer

$x = 4$

8

$2x + 9 = x + 15$

Answer

$x = 6$

9

$5x - 3 = 2x + 12$

Answer

$x = 5$

10

$3(x - 1) = 2(x + 4)$

Answer

$x = 11$

Q1
Solve $x + 7 = 12$
10 XP
Q2
Solve $2x - 5 = 9$
10 XP
Q3
Solve $3(x - 2) = 12$
10 XP
Q4
Solve $5x + 2 = 3x + 10$
10 XP
Q5
Which is the first step to solve $\frac{x}{3} + 4 = 7$?
10 XP
SAQ1
Short Answer
15 XP

Solve $4x + 7 = 23$ and check your answer.

Sample solution

$4x + 7 = 23$. Subtract 7: $4x = 16$. Divide by 4: $x = 4$.

Check: $4(4) + 7 = 16 + 7 = 23$ ✓

SAQ2
Short Answer
15 XP

Solve $2(3x - 1) = 4x + 8$, showing all steps.

Sample solution

Expand: $6x - 2 = 4x + 8$.

Subtract $4x$: $2x - 2 = 8$. Add 2: $2x = 10$. Divide by 2: $x = 5$.

Check: LHS $= 2(15-1) = 28$. RHS $= 20+8 = 28$. ✓

SAQ3
Short Answer
20 XP

Explain why you must do the same operation to both sides of an equation. Use an example.

Sample answer

An equation is like a balanced scale — both sides are equal. If you change only one side, the balance breaks.

For example, in $x + 5 = 12$: subtract 5 from both sides: $x + 5 - 5 = 12 - 5 \Rightarrow x = 7$. Check: $7 + 5 = 12$ ✓

Stretch Challenges
extension

Stretch 1: Solve $\dfrac{2x + 5}{3} = 7$

Solution

Multiply both sides by 3: $2x + 5 = 21$. Subtract 5: $2x = 16$. Divide by 2: $x = 8$.

Check: $\frac{2(8)+5}{3} = \frac{21}{3} = 7$ ✓

Stretch 2: Solve $5(x - 3) = 2(x + 6)$

Solution

Expand: $5x - 15 = 2x + 12$. Subtract $2x$: $3x - 15 = 12$. Add 15: $3x = 27$. Divide by 3: $x = 9$.

Check: LHS $= 5(6) = 30$. RHS $= 2(15) = 30$ ✓

Stretch 3: Three consecutive numbers add to 72. If the middle number is $n$, write an equation and solve for $n$.

Solution

Numbers: $(n-1), n, (n+1)$. Equation: $(n-1) + n + (n+1) = 72$. Simplify: $3n = 72$. So $n = 24$.

The three numbers are 23, 24, 25.

Key Takeaways
copy these
Inverse operations$+$ undone by $-$; $\times$ undone by $\div$. Use them to isolate the unknown.
Balance methodDo the same operation to both sides to keep the equation balanced.
Reverse BODMASUndo addition/subtraction before multiplication/division.
Brackets firstExpand brackets before solving. $3(x-2) = 3x - 6$, not $3x - 2$.
Two-step templateSolve $ax + b = c$: subtract $b$, then divide by $a$. $x = \frac{c-b}{a}$
Always checkSubstitute answer back into the original equation — catches 90% of errors.
Lesson Complete!
+100 XP

You've reviewed all types of linear equations — one-step, two-step, brackets, and unknowns on both sides. You're ready for simultaneous equations!