Cliste School
Get started

Number & Algebra

Linear Relationships: Expanding, Factorising, Equations and Graphs

Year 8

  • ✓By the end of this lesson students will be able to expand algebraic expressions using the distributive law.
  • ✓By the end of this lesson students will be able to factorise algebraic expressions by identifying common factors.
  • ✓By the end of this lesson students will be able to solve linear equations using inverse operations.
  • ✓By the end of this lesson students will be able to construct tables of values and graph linear relationships on the Cartesian plane.

Key concepts

Expanding Algebraic Expressions

Expanding an algebraic expression means removing the grouping symbols (brackets) by multiplying the term outside the bracket by each term inside the bracket. This process uses the distributive law. For example, a(b + c) = ab + ac.

a(b + c) = ab + ac
Factorising Algebraic Expressions

Factorising is the reverse process of expanding. It involves identifying the highest common factor (HCF) of all terms in an expression and writing the expression as a product of the HCF and another expression in brackets. For example, ab + ac = a(b + c).

ab + ac = a(b + c)
Linear Equations

A linear equation is an equation where the highest power of the variable is 1. Solving a linear equation means finding the value of the variable that makes the equation true. This is typically done by using inverse operations to isolate the variable on one side of the equation, ensuring that whatever operation is performed on one side is also performed on the other side to maintain equality.

Graphing Linear Relationships

A linear relationship between two variables (usually x and y) can be represented as a straight line on a Cartesian plane. To graph a linear relationship, we typically create a table of values by choosing several x-values, substituting them into the equation to find the corresponding y-values, and then plotting these ordered pairs (x, y) on the Cartesian plane. Once plotted, a straight line is drawn through these points.

Key facts to remember

  • 1The distributive law states that a(b + c) = ab + ac.
  • 2Factorising is the reverse process of expanding; it involves finding the highest common factor (HCF).
  • 3To solve a linear equation, use inverse operations to isolate the variable, performing the same operation on both sides.
  • 4A linear relationship, when graphed, always forms a straight line.
  • 5The Cartesian plane uses ordered pairs (x, y) to locate points, where x is the horizontal coordinate and y is the vertical coordinate.
  • 6When graphing, it's good practice to calculate at least three points to ensure accuracy and confirm they lie on a straight line.

Worked examples

Example 1

Expand the expression: 4(3x - 7)

IApply the distributive law: multiply 4 by each term inside the bracket.
II4 × 3x - 4 × 7
III12x - 28

Answer

12x - 28

Remember to multiply the number outside the bracket by EVERY term inside the bracket.

Example 2

Factorise the expression: 10y + 15

IIdentify the highest common factor (HCF) of 10y and 15.
IIFactors of 10y: 1, 2, 5, 10, y, 2y, 5y, 10y
IIIFactors of 15: 1, 3, 5, 15
IVThe HCF is 5.
VDivide each term by the HCF and write the expression in factorised form.
VI5(10y/5 + 15/5)
VII5(2y + 3)

Answer

5(2y + 3)

You can check your answer by expanding the factorised expression to see if it returns the original expression.

Example 3

Solve the linear equation: 3x - 8 = 13

ITo isolate the term with 'x', add 8 to both sides of the equation.
II3x - 8 + 8 = 13 + 8
III3x = 21
IVTo isolate 'x', divide both sides by 3.
V3x / 3 = 21 / 3
VIx = 7

Answer

x = 7

Always perform the same operation on both sides of the equation to maintain balance.

Example 4

Graph the linear relationship y = 2x - 1 for x values from -2 to 2.

ICreate a table of values by substituting x-values into the equation y = 2x - 1:
IIWhen x = -2, y = 2(-2) - 1 = -4 - 1 = -5. Point: (-2, -5)
IIIWhen x = -1, y = 2(-1) - 1 = -2 - 1 = -3. Point: (-1, -3)
IVWhen x = 0, y = 2(0) - 1 = 0 - 1 = -1. Point: (0, -1)
VWhen x = 1, y = 2(1) - 1 = 2 - 1 = 1. Point: (1, 1)
VIWhen x = 2, y = 2(2) - 1 = 4 - 1 = 3. Point: (2, 3)
VIIPlot these ordered pairs on a Cartesian plane.
VIIIDraw a straight line through all the plotted points.

Answer

A graph showing the points (-2, -5), (-1, -3), (0, -1), (1, 1), (2, 3) connected by a straight line.

Use a ruler to draw the straight line and label your axes (x and y).

Common mistakes

  • ✗When expanding, forgetting to multiply the term outside the bracket by *every* term inside the bracket.
  • ✗When factorising, not finding the *highest* common factor, leading to an incompletely factorised expression.
  • ✗When solving equations, incorrectly applying inverse operations (e.g., adding when you should subtract, or multiplying when you should divide).
  • ✗Making arithmetic errors when substituting values into equations for graphing.
  • ✗Plotting points incorrectly on the Cartesian plane or not drawing a straight line through all points when graphing.

Exam tips

  • ★Always show all your working steps clearly for expanding, factorising, and solving equations. This allows for partial marks even if the final answer is incorrect.
  • ★Check your solutions for linear equations by substituting your answer back into the original equation to ensure both sides are equal.
  • ★When graphing, use a sharp pencil and a ruler to draw neat and accurate lines. Label your axes (x and y) and clearly mark the scale.
  • ★For factorising, if you're unsure, expand your factorised answer to see if it matches the original expression.

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum