Writing Linear Equations in Slope-Intercept Form

Linear function rules (linear equations) can be written in different ways to highlight different characteristics. In this section, writing these rules in slope-intercept form will be explored.

Method

Writing the Equation of a Line in Slope-Intercept Form using Two Points

To write linear equations in slope-intercept form, $y = m x + b,$ the slope, $m,$ and the $y$ -intercept, $b,$ of the line must be known. When two points on the line are known, the following method can be used.
Write the equation of the line that passes through the point $(- 4, 1)$ and $(8, 4) .$

1

Find the slope

When two points on a line are known, the slope of the line can be found using the slope formula. Here, the coordinates

(8, 4)

and

(- 4, 1)

can be substituted in place of

(x_{1}, y_{1})

and

(x_{2}, y_{2}),

respectively.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePointsSubstitute

(- 4, 1)

&

(8, 4)

m = \frac{1 - 4}{- 4 - 8}

SubTermsSubtract terms

m = \frac{- 3}{- 1 2}

CalcQuotCalculate quotient

m = 0.25

The slope of the line passing through the two points is

m = 0.25 .

2

Replace

m

with the slope

The equation $y = m x + b$ can be re-written with $m = 0.25 .$ This gives $y = 0.25 x + b .$

3

Find

b

using a point

Next, the

y

-intercept can be found by replacing

x

and

y

in the equation with either of the given points. Then, solving for

b

gives the

y

-intercept. Here, the arbitrarily chosen point that will be used is

(8, 4) .

Therefore, substitute

x = 8

and

y = 4

into the equation from Step 2.

y = 0.25 x + b

SubstituteII

x = 8

,

y = 4

4 = 0.25 \cdot 8 + b

MultiplyMultiply

4 = 2 + b

SubEqn

LHS - 2 = RHS - 2

2 = b

RearrangeEqnRearrange equation

b = 2

Thus, the

y

-intercept is

b = 2 .

4

Write the equation

Lastly, the complete equation in slope-intercept form can be written by replacing the value of the $y$ -intercept found above. Here, $b = 2$ will be substituted into the equation from Step 2. This gives $y = 0.25 x + 2 .$

Write the equation of the line that passes through the given points

fullscreen

Exercise

Write the equation of the line that passes through the point $(3, 1)$ and has the same $y$ -intercept as the line $y = 9 x + 4 .$

Show Solution

Solution

A line in slope-intercept form is given by the equation

y = m x + b,

where

m

is the slope of the line and

b

is the

y

-intercept. The line

y = 9 x + 4

has a

y

-intercept of

(0, 4) .

We want our line to have the same

y

-intercept. Therefore, the equation of the new line must also have the value

b = 4 .

This gives

y = m x + 4 .

Our line must also pass through the point

(3, 1) .

We can solve for

m

in the equation above by substituting this point for

x

and

y .

y = m x + 4

SubstituteII

x = 3

,

y = 1

1 = m \cdot 3 + 4

MultiplyMultiply

1 = 3 m + 4

SubEqn

LHS - 4 = RHS - 4

- 3 = 3 m

DivEqn

LHS / 3 = RHS / 3

- 1 = m

RearrangeEqnRearrange equation

m = - 1

The slope of the new line is

m = - 1 .

Thus, we can write the complete equation as

y = - x + 4 .

Method

Writing the Equation of a Line in Slope-Intercept Form from a Graph

To write the equation of the graph of a line in slope-intercept form, $y = m x + b,$

the $y$ -intercept, $b$ , and the slope of the line, $m,$ must be found. The following method can be used. As an example, consider the line shown.

1

Find the

y

-intercept

The $y$ -intercept is the point where the graph intersects the $y$ -axis. From the diagram, it can be seen that the $y$ -intercept is $(0, - 4) .$

2

Replace

b

with the

y

-intercept

The $y$ -coordinate of the $y$ -intercept can be substituted into $y = m x + b$ for $b .$ Here, substituting $b = - 4$ gives $y = m x - 4 .$

3

Find the slope

Next, the slope of the line must be determined. From a graph, the slope of a line can be expressed as $m = \frac{rise}{run},$ where rise gives the vertical distance between two points, and run gives the horizontal distance. To find the slope, use any two points. Here, use the already marked $y$ -intercept and the arbitratily chosen $(2, 2) .$

From the lines drawn, it can be seen that the $rise = 6$ and the $run = 2 .$ Therefore, the slope is $m = \frac{6}{2} = 3 .$

4

Replace

m

with the slope

The complete equation of the line can now be written by substituting the value of $m$ into the equation from Step 2. Here, substitute $m = 3 .$ $y = 3 x - 4 .$

1

2

3

4

Example

Write the equation of the line that passes through the given points

1

2

3

4