The essentials
A linear equation in two variables represents a straight line. A common form is y = mx + b.
The slope m tells us how much y changes when x increases by 1. The y-intercept b tells us where the line crosses the y-axis.
For y = 2x + 1, the slope is 2 and the y-intercept is 1. Move 1 unit to the right and 2 units upwards to find another point on the line.
To draw the graph, choose x-values, calculate y, plot the pairs (x, y), and join them with a straight line. Every point on the line satisfies the equation.
At the y-axis, x = 0, giving the point (0, 1). At the x-axis, y = 0, so 2x + 1 = 0 and x = −0.5. The x-intercept is (−0.5, 0).
Worked example
For y = 2x + 1:
When x = −2, y = −3.
When x = −1, y = −1.
When x = 0, y = 1.
When x = 1, y = 3.
When x = 2, y = 5.
To solve 2x + 1 = 5 graphically, draw the horizontal line y = 5. It meets y = 2x + 1 at (2, 5). Read the x-coordinate: x = 2.
Algebraic check: subtract 1 to get 2x = 4, then divide by 2 to get x = 2.