What is Slope Intercept Form?
The slope intercept form of linear equations is an algebraic representation of straight lines: y = mx + b.
Use this formula to graph a line for two variables using the X and Y axes on the coordinate plane. The slope intercept form uses two pieces of information to produce the line.
- Slope: Defines the angle of the line on a graph, including direction and steepness.
- Intercept: The point where the line crosses the y-axis.
The prominence of these two properties gives this form of linear equation its name. It is the most common way to represent straight lines.
In this post, learn about interpreting the slope intercept form, using it to graph lines, and how to find linear equations when given two points.
Using this linear equation requires understanding the x and y axes on the coordinate plane. To learn more, read X and Y Axes.
Slope Intercept Form Linear Equation
The slope intercept equation uses the following format:
y = mx + b
- m = the slope.
- b = y-intercept.
- x and y pairs are line coordinates (x, y).
In this equation, m and b can be any real numbers.
For example, 3x – 4.
This form allows you to pull important information from the equation quickly. At a glance, you know that the slope is 3, and the y-intercept is -5.
To find points on the line, enter x-values into the linear equation to calculate the corresponding y-values. Together, the (x, y) coordinates are points on the line. For instance, using the above formula, when x = – 1 and x = 2:
- y = 3 * -1 – 4 = -7
- y = 3 * 2 – 4 = 2
Hence, (-1, -7) and (2, 2) are two points on the line. Frequently, you’ll want to graph the slope intercept linear equations to see the lines. We’ll graph these points below.
Here’s how all the components of the slope intercept form equation come together to create a line on a graph.
Consider the previous linear equation: y = 3x – 4.
- m = 3
- b = -4
You can see the points (-1, -7) and (2, 2) that we calculated earlier. I also show the y-intercept point (0, -4). Slope is rise divided by run, producing m = 3.
Linear Equation Examples
You’ll see slope values using fractions or decimals. For instance, the following linear equations are equivalent:
- y = 1/2x – 5
- y = 0.5x – 5
In some cases, you’ll see the order of the terms switched. The equations y = mx + b and y = b + mx are equivalent because the order does not matter for addition (i.e., it is commutative). Hence, the following equations have the same solutions:
- y = -7x + 2
- y = 2 – 7x
When someone provides the slope and intercept for a line, you can easily use those values to create the slope intercept form equation. Or you can start with the linear equations and find the slope and intercept values, as shown below:
|9||-12||y = 9x – 12|
|-2/3||3||y = -2/3x + 3|
|16||4||y = 16x + 4|
Interpreting the Coefficients for Linear Equations
Let’s dig into the coefficients for the slope intercept form a bit more: m and b. These are the slope and y-intercept, respectively.
Slope Coefficient m
The slope defines the angle of the line on the x and y coordinate plane. Because you multiply m*x, the slope coefficient describes the change in y for every one-unit increase of x.
For example, consider the slopes in the following linear equations:
- 4x + 12: Increasing x by 1 makes y increase by 4.
- -3x + 2: A one-unit increase in x causes y to decrease by 3.
You can think of one-unit increases in x as moving right along the x-axis, and the slope intercept form equation tells you how y changes with each shift. Hence, positive slope coefficients cause y-values to increase as you move right. Conversely, negative slopes cause y to decrease.
Consequently, the sign of the coefficient determines whether you have an upward or downward slope as you move right.
The absolute size of the coefficient determines how much y changes for each one-unit shift. Larger absolute values indicate that y changes more dramatically for a one-unit change in x. In other words, larger absolute slopes produce steeper lines.
Y-Intercept Coefficient b
The y-intercept is where the line crosses the y-axis. Unlike m, b does not multiply x, making it a constant in the equation.
The y-intercept point on the line always has an x-value of zero because that falls right on the y-axis. Consequently, when you have an intercept of b, the slope intercept form produces a point on the line at (0, b).
You can solve this by entering an x-value of zero into the slope intercept form equation:
y = m*0 + b = b
y = 9x – 8 produces a Y-intercept at (0, -8)
How to Graph Slope Intercept Form Examples
Using the slope intercept form of a linear equation to graph lines is easy. Simply calculate two (x, y) points by entering two x-values into the formula and finding the corresponding y-values. Then, draw a straight line through the two points.
In fact, it’s even easier than that because you can use m and b to find two points using minimal calculations: (0, b) and (1, b + m).
Here are the step-by-step instructions for graphing a line using the slope intercept form:
- Plot a point at (0, b).
- Move right by one step and plot a point at (1, b + m)
- Draw a straight line through both dots.
Let’s graph the following three lines using the two points for each line from the following linear equations:
- y = 2x + 4: (0, 4) (1, 6)
- y = 0.5x – 2: (0, -2) (1, -1.5)
- y = -1x – 5: (0, -5) (1, -6)
These lines illustrate various properties, including positive and negative slopes and intercepts, and steeper and shallower angles. To create this graph, I found the two points for each line using the steps above. Then, I drew a straight line through the points.
How to Find Slope Intercept Form with Two Points
Suppose you are given two points on a line and need to find the slope intercept form of the linear equation that fits them. How do you do that?
You just need a bit of algebra!
For this example, we’re given the two points of (1, 6) and (3,10).
Find the Slope
First, we need to calculate the slope. Slope is the rise over run. Use the slope formula below:
For this calculation, it doesn’t matter which point is 1 vs. 2. If the math is easier one way, go with it!
I’ll enter the values for our two points: (1, 6) and (3,10) into the slope formula.
So, m = 2.
Find the Intercept
Second, we calculate the intercept. We need to use the m we found before and some algebra.
Here are the steps for finding the intercept:
- Plug the value of the slope (m) into the equation.
- Plug the (x, y) values for one of the points into the equation.
- Solve for b: b = y – mx
It doesn’t matter which point you use. Pick the easier one.
I’ll enter the values for m = 2 and the x and y-values for (3, 10) into y = mx + b and then solve for b.
So, b = 4.
Write the Slope Intercept Form Linear Equation (y = mx + b)
Now, enter the values we found for m and b into y = mx + b.
For our example, m = 2 and b = 4, therefore:
y = 2x + 4
This article looks at the slope intercept form of a linear equation from an algebraic viewpoint. In statistics, we use this form for linear regression. To learn more about that, click the following links:
- Linear Regression Equations
- Interpreting the Slope Coefficients in Regression
- Interpreting the Intercept in Regression