Practice Maths

Solving Linear Equations

Key Ideas

Key Terms

linear equation
Has a variable to the power of 1 (no x2 or higher).
balance method
Whatever you do to one side, do the same to the other side.
Inverse operations
Undo addition with subtraction; undo multiplication with division.
Brackets
Expand first, then solve.
Variables on both sides
Collect variable terms on one side first, constants on the other.
Method Reference
Equation typeStrategy
One-stepApply one inverse operation
Two-stepUndo addition/subtraction first, then multiplication/division
BracketsExpand, then solve as two-step
Variables on both sidesMove all x-terms to one side, constants to the other
Hot Tip When an equation has brackets, expand them before trying to move terms. Trying to “move” a bracket as a whole unit is a very common mistake.

Worked Example

Question: Solve 3(2x − 1) = 15.

Step 1 — Expand the brackets.
6x − 3 = 15

Step 2 — Add 3 to both sides.
6x = 18

Step 3 — Divide both sides by 6.
x = 3

Check: 3(2(3) − 1) = 3(5) = 15 ✓

The Balancing Method

An equation is like a set of balanced scales — both sides have equal value. The balancing method (also called the "do the same to both sides" method) keeps the equation balanced by performing the same operation on both sides until the variable is isolated.

Solve 3x − 7 = 11: add 7 to both sides → 3x = 18 → divide both sides by 3 → x = 6. Check: 3(6) − 7 = 18 − 7 = 11. Correct.

Multi-Step Equations

When equations have multiple operations, work in reverse order of operations: undo addition/subtraction first, then undo multiplication/division. This is the opposite of BODMAS.

Solve 4x + 3 = 19: subtract 3 → 4x = 16 → divide by 4 → x = 4.

Solve 2(3x − 1) + 5 = 21: expand brackets first → 6x − 2 + 5 = 21 → 6x + 3 = 21 → 6x = 18 → x = 3.

Equations with Brackets

When there are brackets, always expand first, then collect like terms, then isolate the variable.

Solve 3(x + 4) = 2(x − 1) + 17: expand → 3x + 12 = 2x − 2 + 17 → 3x + 12 = 2x + 15 → subtract 2x → x + 12 = 15 → x = 3.

Collecting Variables on One Side

When the variable appears on both sides, move all variable terms to one side and all constant terms to the other.

Solve 5x − 3 = 2x + 9: subtract 2x from both sides → 3x − 3 = 9 → add 3 → 3x = 12 → x = 4.

It does not matter which side you collect the variable on — choose the side that keeps the coefficient positive to avoid sign errors.

Always check: Substitute your solution back into the original equation to verify both sides are equal. This takes less than 30 seconds and can save you from losing marks on a solution that looked right but had a small error. Write "Check: LHS = ... = RHS" on your working to show the marker you have verified your answer.

Mastery Practice

  1. Solve each one-step equation. Fluency

     EquationSolution
    (a)x + 5 = 12 
    (b)3x = 21 
    (c)x − 7 = −3 
    (d)x ÷ 4 = 6 
    (e)y + 15 = 7 
    (f)2p = −14 
    (g)m − 9 = 4 
    (h)n ÷ 3 = −5 
    (i)4k = −20 
    (j)a + 8 = −2 

  2. Solve each two-step equation. Fluency

     EquationSolution
    (a)2x + 3 = 11 
    (b)5x − 4 = 16 
    (c)3y + 7 = 1 
    (d)4m − 3 = 25 
    (e)−2n + 9 = 1 
    (f)6p + 4 = −8 
    (g)10 − 3x = 4 
    (h)15 − 2y = −1 
    (i)7a + 2 = −12 
    (j)−3k − 5 = 10 

  3. Solve each equation with variables on both sides. Fluency

     EquationSolution
    (a)3x + 2 = x + 10 
    (b)5x − 3 = 2x + 9 
    (c)7y + 1 = 4y + 13 
    (d)4m − 5 = m + 7 
    (e)2(x + 4) = x + 11 
    (f)3(2x − 1) = 5x + 4 
    (g)6x − 5 = 4(x + 3) 
    (h)8n + 3 = 5n − 6 

  4. True or False? Check each claim by substituting the given value. Write True or False, and if false, find the correct solution. Fluency

    1. x = 4 is a solution of 3x − 5 = 7
    2. x = 2 is a solution of 4x + 7 = 2x + 15
    3. x = −3 is a solution of 5x + 9 = −6
    4. x = 6 is a solution of 2(x − 1) = x + 4
    5. x = 5 is a solution of 3(2x − 4) = 2x + 10
    6. x = 0 is a solution of 7x + 3 = 3

  5. Finding a number. Understanding

    Number puzzles. In each part, write an equation and solve it to find the unknown number.
    1. A number is doubled and then 7 is added. The result is 23. Find the number.
    2. Three consecutive integers add to 75. Let the smallest integer be n. Find all three integers.
    3. A number is multiplied by 5 and then 8 is subtracted. The result equals twice the number plus 7. Find the number.

  6. Age problems. Understanding

    Ages. Define a variable for each unknown age, write an equation, and solve.
    1. Sam is 4 years older than his sister Mia. The sum of their ages is 26. How old is each person?
    2. A mother is three times as old as her daughter. In 10 years, the mother will be twice the daughter’s age. How old are they now?

  7. Perimeter problems. Understanding

    Shapes. Write an equation using the given perimeter, then solve for the unknown dimension.
    1. The perimeter of a rectangle is 50 cm. Its length is 3 cm more than twice its width. Find the width and length.
    2. An equilateral triangle has side length (2x + 1) cm. Its perimeter is 27 cm. Find x and the side length.

  8. Given the solution, find the missing number. Understanding

    Reverse thinking. Each equation has a missing value (shown as k). Use the given solution to find k.
    1. The equation 3x + k = 14 has solution x = 2. Find k.
    2. The equation kx − 5 = 13 has solution x = 4. Find k.
    3. The equation 2(x + k) = 18 has solution x = 5. Find k.

  9. Multi-step word problem. Problem Solving

    School supplies. Maya buys 4 pens and a notebook. The notebook costs $3 more than a pen. The total cost is $18.
    1. Let the cost of a pen be $p. Write an expression for the cost of the notebook.
    2. Write an equation for the total cost and solve for p.
    3. State the cost of the notebook and verify your answer.

  10. Multi-step problem with brackets. Problem Solving

    Fence sections. A farmer builds a fence from three identical sections plus one end post. Each section is 2 m longer than three times the post width. The total length of the fence is 33.5 m, and the post is 0.5 m wide.
    1. Let the section length be s metres. Write an equation for the total fence length.
    2. Solve for s.
    3. Verify that your answer is consistent with the post width given.
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