y = mx + c

Where the line crosses the y-axis

Every straight line that is not vertical has an equation of the form y = mx + c, where m and c are numbers.

Start with y = x + 2. Every point on the y-axis has x = 0, and putting x = 0 into the equation gives y = 0 + 2 = 2. So the line crosses the y-axis at (0, 2). This height is called the y-intercept.

The same happens for any line y = mx + c: when x = 0, the mx term is 0 and y = c. So c is the y-intercept.

The line y = x + 2 crosses the y-axis at (0, 2), so its y-intercept is c = 2.

The gradient is the steepness

Now take y = 2x + 1. When x = 0, y = 1. When x = 1, y = 3. When x = 2, y = 5. Each step of 1 to the right adds 2 to y, because 2x grows by 2 each time x grows by 1.

So the line rises 2 for every 1 across. This number is the gradient of the line: how far it goes up for each 1 it goes across. In y = mx + c, the gradient is m, the number multiplying x. A bigger m makes a steeper line.

From the marked point (1, 3), a run of 1 across comes with a rise of 2 up the line y = 2x + 1: the gradient is 2.

Lines that fall have a negative gradient

In y = −2x + 6, each step of 1 to the right takes 2 away from y: when x = 0, y = 6; when x = 1, y = 4; when x = 2, y = 2. The line falls from left to right, and its gradient is −2.

The gradient is the rise divided by the run, measured left to right. A line that goes down has a negative rise, so its gradient is negative. A level line has no rise at all, so its gradient is 0: the line y = 3 is y = 0x + 3.

The line turns about (0, 2). $\Delta x$Δx is the run and $\Delta y$Δy is the rise, and the gradient is $m=\Delta y/\Delta x$m = Δy / Δx. Swing the point down: the rise becomes negative and so does m. Make the line level and m = 0.

Reading m and c from an equation

In y = 3x − 2, the gradient is 3 and the y-intercept is −2, because y = 3x − 2 is y = 3x + (−2). The line crosses the y-axis at (0, −2) and rises 3 for every 1 across.

Keep the sign with each number. In y = 5 − x, the x term is −x, so the gradient is −1, and the y-intercept is 5.

Read m and c only when y is on its own on one side. 2y = 4x + 6 does not have gradient 4: divide every term by 2 to get y = 2x + 3, and the gradient is 2 and the y-intercept is 3.

One handle sits where the line meets the y-axis and sets c. The other sets the gradient m, which the handle calls the slope, another name for it. Make the line pass through (2, 3): each value of c needs its own gradient to get there.

The equation from two points

A line passes through (1, 5) and (3, 9). From the first point to the second, x goes from 1 to 3, a run of 3 − 1 = 2, and y goes from 5 to 9, a rise of 9 − 5 = 4. The gradient is the rise divided by the run: $m=4/2=2$m = 4 / 2 = 2.

So the line is y = 2x + c, and c is still to be found. The point (1, 5) is on the line, so x = 1 and y = 5 make the equation true: 5 = 2 × 1 + c, which gives c = 3. The line is y = 2x + 3.

Check it with the other point: 2 × 3 + 3 = 9, which is the y-coordinate of (3, 9).

From the marked point (1, 5) to (3, 9) is a run of 2 and a rise of 4, so the gradient is 4 ÷ 2 = 2. The line meets the y-axis at 3.

The usual mistakes

Swapping m and c. In y = 4x + 3 the gradient is 4, the number multiplying x, and the y-intercept is 3, the number added on.

Adding m and c. In y = 4x + 3, the value 7 is y at x = 1, one step along the line. The y-intercept is y at x = 0, which is 3.

Dividing the run by the rise. The gradient is the change in y divided by the change in x: for (1, 5) and (3, 9) it is 4 ÷ 2 = 2, not 2 ÷ 4.

Writing a line from one point and its gradient

Looking ahead: a line can also be written straight from one point and its gradient, in point-slope form. A line with gradient m through the point $\left(x_1,y_1\right)$(x₁, y₁) is $y-y_1=m\left(x-x_1\right)$y − y₁ = m(x − x₁), where $x_1$x₁ and $y_1$y₁ are the coordinates of that point.

The graph of a taxi fare has gradient 1.5 and passes through (2, 7), so its equation is y − 7 = 1.5(x − 2). Expand the bracket: y − 7 = 1.5x − 3. Add 7 to both sides: y = 1.5x + 4.

Substituting the point into y = 1.5x + c gives the same answer: 7 = 1.5 × 2 + c = 3 + c, so c = 4.