Linear Relationships and Graphs

Introduction

When one quantity changes at a steady rate with another, their relationship is linear. Its graph is a straight line. Many real situations are approximately linear: a taxi fare, a phone plan or a plant growing a little each day.

Coordinates

A point is written (x, y): x is the horizontal position and y is the vertical position. The point (3, 5) is 3 along and 5 up.

The equation y = mx + c

A straight line has the form:

y = m x + c

• m is the slope (gradient): how much y changes when x increases by 1.
• c is the y-intercept: the value of y when x is 0, where the line crosses the y-axis.

Example: y = 2x + 1.

Each step of 1 in x adds 2 to y, so the slope is 2. The line crosses the y-axis at 1.

A positive slope rises left to right, a negative slope falls, and a slope of 0 is horizontal.

Slope from two points

slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1).

For the points (1, 3) and (3, 7): slope = (7 - 3) / (3 - 1) = 4 / 2 = 2.

To find c, substitute one point: 3 = 2(1) + c, so c = 1. The line is y = 2x + 1.

Modelling

A phone plan: 10 per month plus 2 per gigabyte. Cost C = 2g + 10. The slope 2 is the price per GB, and the intercept 10 is the fixed fee. Using 8 GB costs 2(8) + 10 = 26.

To find g when C = 30: 2g + 10 = 30, so g = 10.

Remember that models have limits. A line can fit data in a range, but extending it too far may be nonsense. A plant does not keep growing at the same rate forever.

Solving by graph

Two lines meet where both equations are true. For y = 2x + 1 and y = x + 4, set 2x + 1 = x + 4, so x = 3 and y = 7. They cross at (3, 7). This can describe, for instance, when two phone plans cost the same.

Reading graphs

Check axis labels, units and scales. A steep line may only look steep because of the axis scale. Ask what the slope means in the story, with its units, such as "pounds per kilometre".

Try it yourself

Find the equation of the line through (2, 5) and (6, 13). Use it to find y when x = 10. Then write a linear model for a real situation of your own and explain the slope and intercept.

Summary

• A straight line is y = mx + c, with slope m and y-intercept c.
• Slope is change in y divided by change in x, and two points determine a line.
• Linear models describe steady change but only within sensible limits.