Find Slope and y-Intercept from Equation - Process

y = mx + b

The slope-intercept form is a way to describe the equation of a straight line (a linear function). It appears in the form of $y = mx + b$, where $m$ and $b$ are constants and $y$ and $x$ are variables. Scroll to the bottom of this explanation to see some examples of equations in slope intercept form.

In slope-intercept equations, $y$ and $x$ will always appear as $y$ and $x$, because they are variables that describe the position of specific points on the graph. $m$ and $b$ describe features of the function that stay consistent no matter what $x$ and $y$ are.

Be careful: for an equation to be in lope-intercept form, the $y$ must be by itself on its side of the equation. If there is a constant right before the $y,$ it must be divided out so that $y$ is isolated.

Slope

$m$ is the slope of the line. In slope-intercept form, it is the constant right before the $x.$ Slope can be positive or negative, so keep that in mind when you read slope-intercept equations.

The slope of $y = -\frac14x + 6$ is $-\frac14$. We know this because when an equation is in slope-intercept form, the slope $m$ is the constant right before the $x,$ which here is $-\frac{1}{4}.$ Let's do another question!

The slope of $y = -\frac12x + 7$ is $-\frac12$. We know this because when an equation is in slope-intercept form, the slope $m$ is the constant right before the $x,$ which here is $-\frac{1}{2}.$

The slope of $-2y=x-12$ also has a slope of $-\frac{1}{2}.$ The trick to figuring this one out is to realize that this equation does not start out in slope-intercept form. We can get it into slope-intercept form by dividing both sides by $-2$ in order to isolate the $y$. After we do this, we get a new equation: $y=-\frac{1}{2}x+6.$ This has a slope of $-\frac{1}{2}.$

$y$-intercept

$b$ is the $y$-intercept of the line, or where the line crosses the $y$- (vertical) axis. $b$ can be positive or negative (above or below $y = 0$), so be careful to watch the sign.

The $y$-intercept of $y = 5x + 7$ is $7$. We know this because when an equation is in slope-intercept form, the $y$-intercept $b$ is the constant that we add after the expression containing the $x,$ which here is $7.$

Examples:

Here are some more examples of equations in slope-intercept form!

• $y = 34x + 5$
• $y = -0.5x + 70$
• $y = 3x - 9$
• $y = x\quad$ (note: this could be rewritten as $y = 1x + 0$)

(Video) Finding the Equation of a Line in Slope-Intercept Form

by Mahalo Math

Mahalo Math has a good video that explains the slope-intercept form of linear functions and equations.

$\large{\text{Summary}}$

You need a minimum of two points to be able to define a line. So, suppose we have the points: $$(3,-5)\quad\text{and}\quad(6,2)$$

Using these points, we want to write the equation of the line in slope intercept form which looks like: $$y=mx+b$$ Here, $m$ is the slope and $b$ is the $y$-intercept.

Slope-Intercept Form

Step 1: Find the Slope

The first step is to use the two points to find the slope. We plug the points $({\color{#8925AE}{3}},{\color{#8925AE}{-5}})$ and $({\color{#29A603}{6}},{\color{#29A603}{2}})$ into: $$\begin{align} m&=\frac{y_2-y_1}{x_2-x_1}\\ m&=\frac{{\color{#29A603}{2}}-({\color{#8925AE}{-5}})}{{\color{#29A603}{6}}-{\color{#8925AE}{3}}}\\ m&=\frac{7}{3} \end{align}$$

The slope of the line between these points is $\frac73$. We can fill this in. $$y={\color{#8925AE}{\frac73}}x+b$$

Step 2: Find the $y$-intercept

The next step is to find the $y$-intercept. To do this, we want to plug any one of the points on the line into the equation we have so far, $y=\frac73x+b$, and solve for $b$.

Let's plug in the point $(3,-5)$ and see what we get. $$\begin{align} y&=\frac73x+b\\ -5&=\frac73(3)+b\\ -5&=7+b\\ -12&=b \end{align}$$

Our line crosses the $y$-axis at $(0,-12)$. Using this we can complete the equation of our line. $$y=\frac73x-12$$

Word Problem

A parking garage charges $\$b$, plus $\color{#8925AE}{\$m}$ per hours parked. Carrie parks in the garage for $\color{#8925AE}{2}$ hours and pays $\color{#29A603}{\$5}$. The next day, she parks for $\color{#8925AE}{5}$ hours and pays $\color{#29A603}{\$9.50}$. Find $\color{#8925AE}{m}$.

We know the formula for slope is: $$m=\frac{y_2-y_1}{x_2-x_1}$$

We can plug in the given numbers to find the slope. $$\begin{align} m&=\frac{y_2-y_1}{x_2-x_1}\\ m&=\left(\frac{{\color{#29A603}{\$9.50}}-{\color{#29A603}{\$5}}}{{\color{#8925AE}{5}}-{\color{#8925AE}{2}}}\right)\\ m&=\left(\frac{{\color{#29A603}{\$4.50}}}{{\color{#8925AE}{3}}}\right)\\ m&=\color{#8925AE}{1.5} \end{align}$$

So, the slope of the line that represents the parking fee at the garage is $\color{#8925AE}{m=\$1.50}$.

(Video) Graphing a Linear Equation Using Slope

by mathman1024

This video by mathman1024 walks you through reading linear equations for slope and $y$-intercept, as well as using that information to graph them.

$\large{\text{Summary}}$

The slope-intercept form of a line is: $y=mx+b$ where $m$ is the slope and $b$ is the $y$-intercept. The $y$-intercept is always where the line intersects the $y$-axis, and will always appear as $(0,b)$ in coordinate form.

In the first video, mathman1024 looks at the equation: $$y=\frac32x-1$$ This equation is already in slope-intercept form, so we can tell that the slope is $m=\frac32$ and the $y$-intercept is $b=-1$ or $(0,-1)$.

The steps mathman1024 uses to graph the line are:

1. Find the slope, $m$.
2. Find the $y$-intercept, $b$.
3. Plot the $y$-intercept.
4. Use the slope to plot additional points.
5. Draw the line.

(Video) Graphing a Linear Equation Using Slope

by mathman1024

This video works through a problem where the equation isn't in slope-intercept form. The equation is: $$2x+5y=15$$

So, the first step is to transform the equation into slope-intercept form by getting it so that the $y$ is by itself. $$\begin{align} 2x+5y &=15\\ 2x\,{\color{#8925AE}{-\,2x}}+5y &=15\,{\color{#8925AE}{-\,2x}}\\ 5y &=15-2x\\ \frac{5y}{{\color{#8925AE}{5}}} &=\frac{15-2x}{\color{#8925AE}{5}}\\ y &=3-\frac25x\\ y &=-\frac25x+3 \end{align}$$

Now we can see that the slope is $m=-\frac25$ and the $y$-intercept is the point $(0,3)$. Using the same steps from above the graph becomes:

Practice Problem 2

The line graphed above is $y=x-2$. The slope is $1$ and the $y$-intercept is $-2$.

(Video) Writing Equations of Lines

by mathman1024

Mathman1024 demonstrates how to write linear equations in slope-intercept form in these two videos.

$\large{\text{Summary}}$

The equation for slope-intercept form is: $$y=mx+b$$ In this form, $m$ is the slope of the line and the $b$ is the $y$-intercept.

If you already know what the slope and $y$-intercept of a line are, all you have to do is plug them into the format. Currently, we know: $$\begin{align} m&=7 \\ y\text{-int}&=\left(0,\frac45\right) \end{align}$$ Now, plug in the information into the slope-intercept formula. The equation of this line would be: $$y=7x+\frac45$$

Finding the $y$-intercept

This video works with problems where they give the slope and just one point, not necessarily the $y$-intercept. In these cases, you'll have to use what you know to find the $y$-intercept yourself.

Let's try this with $m=4$ and point $(-3,-2)$, just like in the video. To find the $y$-intercept, we take the slope intercept formula and plug in the slope and the values for $x$ and $y$, then solve the equation for $b$.

$$\begin{align} y&=mx+b\\ \color{#8925AE}{-2}&=({\color{#8925AE}{4}})({\color{#8925AE}{-3}})+b\\ -2&=-12+b\\ -2\,\color{#8925AE}{+\,12}&=-12\,{\color{#8925AE}{+\,12}}+b\\ 10&=b \end{align}$$

And there it is, our $y$-intercept is $10$. Now that we know the slope and the $y$-intercept we can write the equation of the line: $$y=4x+10$$