How to write a linear equation in slope intercept form

The slope intercept form of a linear equation has the following form where the equation is solved for y in terms of x:

y = a + bx

b  is the slope

a  is a constant term. It is the y intercept, the place where the line crosses the y axis.

Example 1

y = -13 + 7x

This equation is in slope intercept form. The y intercept is (0,-13) and the slope is 7.

Example 2

4x + 3y = 12

Rewrite this equation in slope intercept form.

3y = 12 - 4x

The equation is now in slope intercept form. The y intercept is (0,4) and the slope is
-4/3.

Example 3

5x - 3y - 15 = 0

Rewrite this equation in slope intercept form.

3y = -15 + 5x

The equation is now in slope intercept form. The y intercept is (0,-5) and the slope is
5/3.

Example 4

The equation is now in slope intercept form. The y intercept is (0,-7.5) and the slope is
1.5.

[Index]

The general formula for slope-intercept form is , where represents the slope of the line, and represents the -value of the line’s -intercept.

The slope-intercept form of a linear equation makes it easier for us to identify how steep a line is and where it crosses the -axis.

✨ Drag the points on the graph to see how they affect the equation of the line! ✨

When we're given , we first need to find the slope. Then, we can use the slope and one of the given points to solve for the -value of the -intercept and write the equation in slope-intercept form.

What is Standard Form?

The general formula for the standard form of a linear equation is , where , , and are all integers.

We can go from standard form to slope-intercept form by isolating and simplifying:

What is Point-Slope Form?

The general formula for the point-slope form of a linear equation is , where represents the slope of a line that contains the point (, ).

We can go from point-slope form to slope-intercept form by isolating and simplifying:

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Let's first quickly review slope intercept form.

Equations that are written in slope intercept form are the easiest to graph and easiest to write given the proper information.

All you need to know is the slope (rate) and the y-intercept. Continue reading for a couple of examples!

Example 1: Writing an Equation Given the Slope and Y-Intercept

Write the equation for a line that has a slope of -2 and y-intercept of 5.

NOTES: I substituted the value for the slope (-2) for m and the value for the y-intercept (5) for b. The variables x and y should always remain variables when writing a linear equation.

In the example above, you were given the slope and y-intercept. Now let's look at a graph and write an equation based on the linear graph.

Example 2: Writing An Equation Based on a Graph

Write an equation that represents the following graph.

Solution

Step 1: Locate the y-intercept.

Step 2: Locate another point that lies on the line.

Step 3: Calculate the slope from the y-intercept to the second point.

Step 4: Write an equation in slope intercept form given the slope and y-intercept.

Slope = 3

y-intercept = -2

y = mx + b

y = 3x - 2 is the equation that represents this graph.

Note:  You can also check your equation by analyzing the graph. You have a positive slope. Is your graph rising from left to right?

Yes, it is rising; therefore, your slope should be positive!

We've now seen an example of a problem where you are given the slope and y-intercept (Example 1). Example 2 demonstrates how to write an equation based on a graph.

Let's look at one more example where we are given a real world problem. How do we write an equation for a real world problem in slope intercept form?

What will we look for in the problem?

Real World Problems

When you have a real world problem, there are two things that you want to look for!

1. Rate: The rate is your slope in the problem. The following are examples of a rate:

• $3 per day
• $2 an hour
• $5 per person
• $6 a minute

This number is always related to the x value.

"Per" is a key word that is often associated with slope or a rate.

2.  A Flat Fee: A flat fee is your y-intercept. This value is a constant or fixed amount. It never changes!

Use the chart below to help you organize your information as you analyze each word problem. This will help you when writing equations

Take a look at the examples below to better clarify how this chart can help you.

Example 3: Writing Equations for Real World Problems

You are visiting Baltimore, MD. A taxi company charges a flat fee of $3.00 plus an additional $0.75 per mile. Write an equation that you could use to find the cost of a taxi ride in Baltimore, MD. Let x represent the number of miles and y represent the total cost.

• How much would a taxi ride for 8 miles cost?

Solution

The y-intercept is 3. Since there is a flat fee of $3, this value becomes the y-intercept. It is a constant, a value that never changes.

The slope is 0.75. This is the rate per mile. A rate is also the slope.

Therefore, the equation that represents this problem is y = 0.75x + 3

• How much would a taxi ride for 8 miles cost?

In order to determine this cost, we will need to use our equation and substitute 8 for x.

y = 0.75x + 3

y = 0.75(8) + 3

9 = (0.75)(8) + 3

The cost of an 8 mile taxi ride is $9.

Hopefully you now have the hang of writing equations in slope intercept form.

Remember to always look for the slope and the y-intercept.

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