Linear relationships

The number plane

The Cartesian plane (number plane) is built from two axes: the horizontal $x$-axis and the vertical $y$-axis, crossing at the origin $(0, 0)$, usually labelled O. Both axes carry arrows and continue into negative numbers.

The axes divide the plane into four quadrants, numbered 1 to 4 anticlockwise starting from the top right. When you draw a plane yourself, remember: arrows on both axes, labels $x$ and $y$, the origin marked, and an evenly spaced scale.

Plotting points

Every point is written as coordinates $(x, y)$: move across first, then up or down. For $(3, 2)$ go 3 right and 2 up; for $(-2, -4)$ go 2 left and 4 down.

• If $x = 0$ the point lies on the $y$-axis, like $(0, 4)$.
• If $y = 0$ the point lies on the $x$-axis, like $(-1, 0)$.

Tables of values

A linear relationship links $x$ and $y$ with a rule such as $y = 2x - 1$. A table of values substitutes several $x$ values into the rule:

For $x = -1, 0, 1, 2$ the rule $y = 2x - 1$ gives $y = -3, -1, 1, 3$.

Take care with negatives: when $x = -1$, $y = 2 \times (-1) - 1 = -3$, not $1$.

Graphing linear rules

Plot each pair from the table as a point. For a linear rule the points always fall in a straight line — that is what “linear” means. Join them with a ruler, extend the line past your points, and label it with its rule.

The line crosses the $x$-axis where $y = 0$: read it from the graph, or solve the equation. For $y = 2x - 1$, solving $0 = 2x - 1$ gives $x = \frac{1}{2}$.

If a point’s coordinates make the rule true, the point lies on the line: $(3, 5)$ lies on $y = 2x - 1$ because $2 \times 3 - 1 = 5$.

Finding the rule

Stick and matchstick patterns hide linear rules. Count how much the total grows at each step — that number multiplies the term. Then adjust with a constant so the first term works.

Triangles built in a row from matches: one triangle uses 3, and each extra triangle adds 2. The rule is $M = 2T + 1$ (check: $T = 1$ gives $3$). The rule’s power is reaching far terms without drawing: $T = 50$ gives $M = 101$.