Slope Intercept Form

We have seen here when graphing linear relationships that the equation for a straight line can be given in the form y = mx + b. This form is known as the Slope Intercept Form and it is a most useful form as it immediately shows two important things about any straight line when graphed on a Cartesian plane; the slope m, and the y-intercept b.

There are other forms of the equation of a straight line and the examples below will show how to convert from these to the slope intercept form.

There is more here on the slope of a line so we will start by looking at the y-intercept.

The y-intercept

Staight line on a Cartesian grid with point of intersection with y-axis highlighted.

The y-intercept is the point at which a straight line intersects the y-axis. At this intersection point the value of x is always 0 so the y-value can be found algebraically simply by substituting 0 for x in the equation that represents the line as the example below shows.

Move that intercept!

Try the graph generator with different values for the y-intercept (and the slope too if you wish) to see the effect on the line.

Your browser does not support the HTML 5 Canvas.

Enter slope(m) and y-intercept(b) below then click Draw Line


Drawing equation y = 2x – 1

Click Draw Line to graph the equation

Finding the y-intercept

As shown earlier, finding the y-intercept is straightforward if the equation of the line is given in slope intercept form. e.g. for a line with equation y = 3x – 7, the y-intercept is at point (0, -7). If the equation is given in a different form then it can require additional steps as the two examples below show:

Example #1

Find the y-intercept for line with equation
3x + 4y = 12
substitute 0 for x(3 x 0) + 4y = 12
4y = 12
Divide both sides by 4 to isolate y4y ÷ 4 = 12 ÷ 4
y = 3
y-intercept is at point (0,3)

Example #2

Find the y-intercept for line with equation
5x + 7y = -14
substitute 0 for x(5 x 0) + 7y = -14
7y = -14
Divide both sides by 7 to isolate y7y ÷ 7= -14 ÷ 7
y = -2
y-intercept is at point (0,-2)

Slope Intercept Form

The equation of a straight line can be given in different forms. The form y = mx + b is the most common and is known as the Slope Intercept Form. It is not the only form though; for example the equation ax + by = c is shown in what is known as standard form.

Two benefits of the slope intercept form is that both the slope (m) and the y-intercept (b ) are immediately obvious. Let us convert the example above from standard form to slope intercept form:

Converting to Slope Intercept Form

Example #1

Convert 3x + 4y = 12
into Slope Intercept Form
subtract 3x from both sides3x – 3x + 4y = -3x + 12
4y = -3x + 12
Divide both sides by 4 to isolate y4y ÷ 4= (-3x ÷ 4) + (12 ÷ 4)
y = (-3/4)x + 3
y
 = -0.75x + 3

Example #2

Convert -5x + 2y = 15
into Slope Intercept Form
add 5x to both sides5x – 5x + 2y = 5x + 15
2y = 5x + 15
Divide both sides by 2 to isolate y2y ÷ 2= (5x ÷ 2) + (15 ÷ 2)
y = (5/2)x + 7.5
y = 2.5x + 7.5

Solving Other Problems

What is the equation of a line that passes through point (5,6) and has a slope of 3?
Substitute (5,6) for x and y and 3 (slope) for m in the equation in slope intercept form (y = mx + b)6 = (3×5) + b
6 – 15 = b
b = -9
Use the values of m and b to write the equationy = 3x – 9

Worksheets

Use the worksheet(s) below for practice.