Gradient–Intercept Form, Horizontal & Vertical Lines

Any straight line can be written as y = mx + c, and this form is worth memorising because it hands you two facts for free. The m — the number multiplying x — is the gradient, telling you how steep the line is. The c — the number sitting on its own — is the y-intercept, the point (0, c) where the line crosses the y-axis. So y = 3x + 1 has gradient 3 and crosses at (0, 1), while y = −2x + 5 has gradient −2, which is downhill, and crosses at (0, 5). You can describe an entire line without plotting a single point, which is why this form appears constantly.

Lines are not always handed to you in gradient-intercept form. Given 2x + y = 6, you first make y the subject: subtract 2x from both sides to get y = 6 − 2x, then reorder it as y = −2x + 6. Now the gradient is −2 and the y-intercept is 6. That reordering step matters more than it looks. y = 6 − 2x and y = −2x + 6 are the same line, but glancing at the first version and calling the gradient 6 is the classic trap — you have grabbed the constant instead of the coefficient of x. Always rewrite with the x term first, then read off m and c.

Gradient-intercept form lets you sketch a line in two moves. Mark c on the y-axis — for y = 2x − 1 that is (0, −1). Then use m as rise over run to step to a second point: m = 2 means 2 over 1, so go 1 across and 2 up. Rule the line through both. A whole-number gradient is really that number over 1, so m = −3 means 1 across and 3 down. A fractional gradient is already in rise-over-run shape: m = two-thirds means 3 across and 2 up. This is much faster than a table and it is what 'sketch the line' expects in an exam.

Two lines do not look like the rest. The equation y = 3 describes every point whose y value is 3, with x free to be anything — that is a flat, horizontal line at height 3, with gradient zero. The equation x = 2 describes every point whose x value is 2, with y free — that is an upright, vertical line, and its gradient is undefined. These get swapped constantly, because y = 3 mentions y and people expect something tall. The reliable trick is that the equation names the coordinate that never changes: y = 3 pins the height, so the line is flat, and x = 2 pins the across-position, so the line is upright. Note that a vertical line cannot be written as y = mx + c at all.

The final skill: given two points, find the equation. Do the gradient first. From (2, 7) to (5, 16) the gradient is 16 minus 7 over 5 minus 2, which is 9 over 3, so m = 3 and the line is y = 3x + c. Now substitute either point to find c. Using (2, 7): 7 = 3 times 2 plus c, so 7 = 6 + c and c = 1. The equation is y = 3x + 1. Then check with the other point — 3 times 5 plus 1 is 16 ✓. That check has to use the point you did not substitute, because the one you used will always work no matter what mistake you made in the gradient.