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.

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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


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 y

4y ÷ 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 y

7y ÷ 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 sides

3x - 3x + 4y = -3x + 12

4y = -3x + 12

Divide both sides by 4 to isolate y

4y ÷ 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 sides

5x - 5x + 2y = 5x + 15

2y = 5x + 15

Divide both sides by 2 to isolate y

2y ÷ 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 = (3x5) + b

6 - 15 = b

b = -9

Use the values of m and b to write the equation

y = 3x - 9

Worksheets

Use the worksheet(s) below for practice.

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