Another important characteristic is the plot slope that is numerically equal to the coefficient a. In the steps below, we consider calculation of the y-intercept in two common cases. In the first case, a linear equation is defined with a slope and a point with coordinates x1 and y1. In the second one, the equation is given in the form above, and the coefficients a and b are known.

Write the linear equation for the point with coordinates x1 and y1:

y1 = ax1 + b

As an example, consider the linear equation plot with the slope 6 that passes through the point having the following coordinates: x1 = 2 and y1 = 7. The equation for this point is:

7 = 2a + b.

Write the equation for the y-intercept point. Such a point has the coordinate x equals to 0:

y-intercept = 0a + b, or y-intercept = b

Subtract the equation from Step 1 from the equation for the y-intercept point:

y-intercept = b y1 = ax1 + b ------------------- y-intercept - y1 = -ax1.

Then add y1 to both sides of this equation to obtain y-intercept = y1 - ax1 = y1 - (slope * x1). Note that the slope equals to the coefficient a. In our example (see Step 1):

y-intercept = 7 - 6 * 2 = 7 - 12 = -5

Calculate the y-intercept if the coefficients a and b are explicitly defined (the second case). Using the equation from Step 2, you can conclude that the y-intercept equals to the coefficient b:

y-intercept = b

If, for example, the linear equation is given as y = 10x + 3, then the y-intercept is 3.