Year 10 Maths · Unit 2 · Lesson 8

Solving Linear Equations

Master the balance rule, reverse BIDMAS, and handle variables on both sides with confidence.

35 min 8 cards 120 XP
2x + 5 17
Think First
warm-up

Before we begin: What number could replace the box so that $2 \times \square + 5 = 17$ is true? Write down your thinking and explain how you found your answer.

Record your answer in your workbook.
1
The Big Idea
+5 XP to read

A linear equation is a statement that two expressions are equal, where the highest power of the variable is 1. Solving means finding the value of the variable that makes the statement true. The key principle: whatever you do to one side, you must do to the other.

Balance means both sides stay equal. Inverse operations undo each other. Check by substituting back.

x + 3 10
$x + 3 = 10$ → $x = 7$
2
Lesson Objectives
what you will learn
  • Solve one-step equations using inverse operations
  • Solve two-step equations by reversing BIDMAS
  • Expand brackets before isolating the variable
  • Handle variables on both sides of the equation
  • Check solutions by substitution
3
Key Vocabulary
terms to know
Linear equationAn equation where the highest power of the variable is 1, e.g. $2x + 3 = 11$.
SolutionThe value of the variable that makes the equation true.
Inverse operationThe operation that undoes another: addition/subtraction, multiplication/division.
IsolateTo get the variable alone on one side of the equation.
CoefficientThe number in front of a variable. In $5x$, the coefficient is 5.
Constant termA number without a variable. In $3x + 7$, the constant is 7.
4
Spot the Trap
heads-up

Wrong: Moving a term to the other side without changing its sign. $x + 5 = 12$ becomes $x = 12 + 5$.

Right: When moving a term across the equals sign, change its operation: $x + 5 = 12$ becomes $x = 12 - 5$.

Wrong: Dividing before dealing with addition. In $2x + 6 = 14$, dividing by 2 first gives $x + 6 = 7$.

Right: Apply inverse operations in reverse order of operations (BIDMAS backwards): subtract 6 first, then divide by 2.

5
The Balance Rule
+5 XP

Whatever you do to one side of an equation, you must do to the other. Add 5 to both sides. Multiply both sides by 3. Subtract x from both sides. The equation stays balanced, and the solution stays the same.

x + 5 = 12: subtract 5 from both sides. x = 7.

x + 5 = 12 x + 5 - 5 = 12 - 5 x = 7
same op → both sides
Show your steps
Write what you do to both sides on a separate line. It earns marks and catches errors.
Inverse operations
Undo addition with subtraction, multiplication with division, and vice versa.
Check every time
Substitute back: 7 + 5 = 12. If it matches, your solution is correct.
6
Reverse BIDMAS
+5 XP

To isolate a variable, undo operations in the reverse order they were applied. Addition is undone first (subtraction), then multiplication (division), then powers (roots). Reverse BIDMAS keeps your steps safe.

2x + 6 = 14: undo +6 first, then times 2.

2x + 6 = 14 2x = 8 x = 4
inverse order
Write the operation chain
For 3x - 7 = 11, the chain is: x --x3-> 3x --minus 7-> 11. Reverse it.
Check it backwards
Plug x = 4 back in: 2(4) + 6 = 14. If it works, your steps are correct.
One step at a time
Each line of working should perform exactly one inverse operation. Do not rush.
7
Brackets First
+5 XP

When an equation contains brackets, expand them first. Use the distributive law to multiply every term inside the bracket by the factor outside. Only then collect like terms and isolate the variable.

3(2x - 4) becomes 6x - 12. Distribute first, then solve.

3(2x - 4) = 18 6x - 12 = 18 x = 5
distribute, then solve
Watch the sign
A negative outside the bracket flips every sign inside: -2(x - 3) = -2x + 6.
Multiply every term
Do not skip the constant: 3(2x - 4) means 3 times 2x AND 3 times -4.
Simplify after expanding
Combine like terms on each side before trying to isolate x.
8
Variables to One Side
+5 XP

When the variable appears on both sides, move all variable terms to one side and all constants to the other. Subtract the smaller variable term to avoid negatives if possible. Then isolate as usual.

7x + 5 = 3x + 21: subtract 3x, then 5, then divide.

7x + 5 = 3x + 21 4x + 5 = 21 4x = 16 x = 4
subtract smaller term
Subtract the smaller
To avoid negatives, subtract the smaller coefficient: 3x from 7x, not the other way.
Keep signs straight
Moving a term across the equals sign changes its sign: +5 becomes -5, -3x becomes +3x.
Simplify first
If there are brackets, expand them before gathering variable terms to one side.

Watch Me Solve It · 3 examples

Watch Me Solve It · One-step equation
+15 XP per step
Q1
PROBLEM
Solve $\frac{x}{4} = 7$.
  1. 1
    Identify the operation on x
    $\frac{x}{4} = 7$ means x has been divided by 4.
    To undo division by 4, multiply both sides by 4.
  2. 2
    Multiply both sides by 4
    $4 \times \frac{x}{4} = 4 \times 7$
    The 4 on the left cancels with the denominator, leaving x alone.
  3. 3
    Simplify and check
    $x = 28$
    Check: $\frac{28}{4} = 7$. Correct.
Answer$x = 28$
Watch Me Solve It · Two-step equation
+15 XP per step
Q2
PROBLEM
Solve $5x - 8 = 27$.
  1. 1
    Write the operation chain
    $x \xrightarrow{\times 5} 5x \xrightarrow{-8} 27$
    To isolate x, reverse the chain: undo -8 first, then undo times 5.
  2. 2
    Add 8 to both sides
    $5x - 8 + 8 = 27 + 8$
    The -8 and +8 cancel on the left, leaving 5x alone.
  3. 3
    Divide both sides by 5
    $\frac{5x}{5} = \frac{35}{5}$
    Undo multiplication by 5 with division by 5.
  4. 4
    Simplify and check
    $x = 7$
    Check: $5(7) - 8 = 35 - 8 = 27$. Correct.
Answer$x = 7$
Watch Me Solve It · Brackets and both sides
+15 XP per step
Q3
PROBLEM
Solve $3(2x - 4) = 2(x + 5)$.
  1. 1
    Expand both sides
    $3(2x - 4) = 6x - 12$    and    $2(x + 5) = 2x + 10$
    Distribute the 3 and the 2 across every term inside each bracket.
  2. 2
    Write the expanded equation
    $6x - 12 = 2x + 10$
    Now we have variables on both sides. Move them to one side.
  3. 3
    Subtract 2x from both sides
    $6x - 2x - 12 = 2x - 2x + 10$
    Subtracting the smaller coefficient avoids negatives.
  4. 4
    Add 12 to both sides
    $4x - 12 + 12 = 10 + 12$
    Move the constant to the right by adding 12 to both sides.
  5. 5
    Divide by 4 and check
    $\frac{4x}{4} = \frac{22}{4}$    so    $x = 5.5$
    Check: LHS = $3(11 - 4) = 3(7) = 21$. RHS = $2(5.5 + 5) = 2(10.5) = 21$. Correct.
Answer$x = 5.5$

Brain Trainer · 4 problems

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Brain Trainer · Linear Equations
4 problems

Four equations ranging from one-step to variables on both sides. Show each step, then reveal the answer to check your working.

  1. 1 Solve $x + 7 = 15$.

    Subtract 7 from both sides: $x = 8$. Check: $8 + 7 = 15$.

  2. 2 Solve $3x - 5 = 16$.

    Add 5: $3x = 21$. Divide by 3: $x = 7$. Check: $3(7) - 5 = 21 - 5 = 16$.

  3. 3 Solve $2(x + 4) = 18$.

    Divide by 2: $x + 4 = 9$. Subtract 4: $x = 5$. Check: $2(5 + 4) = 2(9) = 18$.

  4. 4 Solve $5x + 3 = 2x + 12$.

    Subtract $2x$: $3x + 3 = 12$. Subtract 3: $3x = 9$. Divide by 3: $x = 3$. Check: $5(3) + 3 = 18$ and $2(3) + 12 = 18$.
Complete in your workbook.

Quick Check · 5 questions

1
Solve $2x + 5 = 17$.
+10 XP
2
Solve $\frac{x}{3} = 9$.
+10 XP
3
Solve $4(x - 2) = 20$.
+10 XP
4
Solve $3x + 7 = x + 15$.
+10 XP
5
A triangle has sides $x$, $(x + 3)$ and $(2x - 1)$ centimetres. Its perimeter is 22 cm. Find $x$.
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Medium 3 MARKS

Q6. Solve $2x + 8 = 5x - 4$. Show all steps and check your answer.

Answer in your workbook.
Apply Medium 3 MARKS

Q7. Ava is three times as old as her brother Ben. In four years, Ava will be twice as old as Ben. Let Ben's current age be $b$ years.

(a) Write an equation for this situation. (1 mark)

(b) Solve the equation to find Ben's current age. (2 marks)

Answer in your workbook.
Analyse Medium 3 MARKS

Q8. Solve $3(2x - 1) = 2(x + 5)$. Show all steps including expansion, collection and checking.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B -- Subtract 5: $2x = 12$. Divide by 2: $x = 6$.

2. C -- Multiply by 3: $x = 27$.

3. A -- Divide by 4: $x - 2 = 5$. Add 2: $x = 7$.

4. B -- Subtract $x$: $2x + 7 = 15$. Subtract 7: $2x = 8$. Divide by 2: $x = 4$.

5. C -- $x + (x + 3) + (2x - 1) = 22$. $4x + 2 = 22$. $4x = 20$. $x = 5$.

Show Your Working Model Answers

Q6 (3 marks): Subtract $2x$: $8 = 3x - 4$ [1]. Add 4: $12 = 3x$ [1]. Divide by 3: $x = 4$ [0.5]. Check: $2(4) + 8 = 16$ and $5(4) - 4 = 16$ [0.5].

Q7 (3 marks): (a) $3b + 4 = 2(b + 4)$ [1]. (b) $3b + 4 = 2b + 8$. Subtract $2b$: $b + 4 = 8$. Subtract 4: $b = 4$ [2]. Ben is 4 years old.

Q8 (3 marks): Expand: $6x - 3 = 2x + 10$ [1]. Subtract $2x$: $4x - 3 = 10$ [0.5]. Add 3: $4x = 13$ [0.5]. Divide by 4: $x = 3.25$ [0.5]. Check: LHS = $3(5.5) = 16.5$, RHS = $2(8.25) = 16.5$ [0.5].

Stretch Challenge · +25 XP, +10 coins

The Fraction Maze

Solve $\frac{2x + 1}{3} = \frac{x - 4}{2}$. Hint: cross-multiply to eliminate the fractions first, then solve the resulting linear equation. Show every step.

Reveal solution

Cross-multiply: $2(2x + 1) = 3(x - 4)$.

Expand: $4x + 2 = 3x - 12$.

Subtract $3x$: $x + 2 = -12$.

Subtract 2: $x = -14$.

Check: LHS = $(2(-14)+1)/3 = (-27)/3 = -9$. RHS = $(-14-4)/2 = -18/2 = -9$. Both match.

R
Quick Review

Balance Rule

Whatever you do to one side, do to the other

Reverse BIDMAS

Undo operations in the opposite order they were applied

Brackets First

Expand before collecting or isolating variables

Variables to One Side

Subtract the smaller coefficient to avoid negatives

One Step Per Line

Clear working earns method marks

Always Check

Substitute your answer back into the original equation

Interactive: Equation Solver

Practise solving linear equations step by step with instant feedback. Try one-step, two-step and multi-step equations.

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