Slope Intercept Form – Equation of a Line Example 1

Slope intercept form is a function of a straight line  with the form

y = mx + b
where
m is the slope of the line and
b is the y-intercept.

Slope refers to how steep the line changes as the values of x change where the y-intercept is where the line crosses the y-axis.

Here are a pair of worked examples to show how to find the equation of a line from different initial conditions.

Slope Intercept Form Example 1

This example shows how to find the equation of the line when given the slope and one point on the line.

Q: What is the Slope intercept form of the line with slope ²⁄₃ and passes through the point (-3, -4)

Here’s the graph of that line. A slope of ²⁄₃ means the line will rise 2 points for every 3 points of x to the right. If we start at the point (-3, -4) we can count 2 up and three over to see the way the line will progress.

The slope intercept form is

y = mx + b

To solve this, we need to know both the slope and the y-intercept. We know the slope, so we need to find the value of b. We know from the given:
m = ²⁄₃
x = -3
y = -4

Plug these into the equation

-4 = ²⁄₃ (-3) + b
-4 = -2 + b
-2 = b

Now we have all we need for the equation of the line.

y = ²⁄₃ x + (-2)
y = ²⁄₃ x – 2

We can see from the graph the line crosses the y-axis at -2 but it nice to have a second opinion.

Slope Intercept Form Example 1 – Alternate Method

There is an alternate method to solve this problem where you can memorize an equation to eliminate the first couple steps. That formula is

(y – y₁) = m(x – x₁)

where x₁ and y₁ are the coordinates of the given point.

Let’s plug in the point (-3, -4) from above.

(y – y₁) = m(x – x₁)
(y – (-4)) = ²⁄₃(x – (-3))
y + 4 = ²⁄₃(x + 3)
y + 4 = ²⁄₃x + ²⁄₃(3)
y + 4 = ²⁄₃x + 2
y = ²⁄₃x + 2 – 4
y = ²⁄₃x – 2

Slope Intercept Form Example 2

The second example, two points of the line are given in the initial conditions.
Let’s find the same line as before. The red line passes through points (-6, -6) and (9, 4). Find the equation of the line.

First we need to find the slope.

The formula to find slope between two points (x₁, y₁) and (x₂, y₂) is

For our two points:
x₁ = -6
y₁ = -6
x₂ = 9
y₂ = 4

Plug these into the formula

m = ¹⁰⁄₁₅
reduce the fraction by factoring out a 5
m = ²⁄₃

Now we can find the y-intercept in the same way we did above. Choose one of the points for x and y. Let’s try the (-6, -6) point.

y = mx + b
-6 = ²⁄₃(-6) + b
-6 = –¹²⁄₃ + b
-6 = -4 + b
-2 = b

Just to show it doesn’t matter which point you choose, let’s use the (9, 4) point.

y = mx + b
4 = ²⁄₃ (9) + b
4 = ¹⁸⁄₃ + b
4 = 6 + b
-2 = b

Plug into the slope intercept formula

y = ²⁄₃ x + (-2)
y = ²⁄₃ x – 2

Which matches what we expected to see from the first example.

The key to this type of problem is to find the slope of the line. Once you have the slope, finding the y-intercept is easy. For more examples of finding the slope, check out the What is Slope? page and examples