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The slope of a line is a crucial concept that we'll need throughout our study of mathematics. In essence, it's a way to quantify the steepness of a line. If we're putting a roof on a house, for example, there are several reasons why we might want to know how steep it is. This could also apply to staircases and roads. When solving math problems, we will often encounter situations when we need to calculate the slope of linear equations. Before we examine these types of problems, it might be valuable to have a look at some related topics, such as y-intercepts and parallel lines and slopes.

In order to calculate the slope of linear equations, we have a simple formula. If we're given two points on a line, defined as $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ , we can define the slope m as follows:

$m=\frac{\left({y}_{2}-{y}_{1}\right)}{\left({x}_{2}-{x}_{1}\right)}$

Therefore, we may be given a problem that goes something like this:

Find the slope of a line that goes through points $\left(4,8\right)$ and $\left(2,2\right)$ .

We can match up these coordinates with our formula:

${x}_{1}=4$

${y}_{1}=8$

${x}_{2}=2$

${y}_{2}=2$

Now it's simply a matter of plugging everything into our original formula:

$m=\frac{\left({y}_{2}-{y}_{1}\right)}{\left({x}_{2}-{x}_{1}\right)}$

$=\frac{\left(2-8\right)}{\left(2-4\right)}$

$=\frac{-6}{-2}$

$=3$

Therefore, our slope is 3.

Graphs with a positive slope will ascend as we go from left to right across the graph. If the line descends from left to right, the slope will be negative.

It's helpful to look at a graph to get a more intuitive feel for what slope is. Consider the image below.

Here we have a line that runs through the points $\left(4,0\right)$ and $\left(-2,-2\right)$ .

First, we draw a vertical line that extends downward from the highest point:

Next, we draw a horizontal line that connects with the remaining point. This forms a slope triangle.

The vertical line can be referred to as the rise and the horizontal line can be referred to as the run. One way to think about slope is the rise divided by the run. In mathematical terms:

$m=\frac{\mathrm{rise}}{\mathrm{run}}$

In this case, we have a rise of 2 and a run of 6.

Therefore, our slope is $\frac{2}{6}$ , which reduces to $\frac{1}{3}$ .

There are two special cases to consider. The first is a horizontal line. Horizontal lines have a run but no rise. Consider our formula.

$m=\frac{\mathrm{rise}}{\mathrm{run}}$

Here, the numerator has a value of zero. Zero divided by anything will always yield a value of zero. Therefore, horizontal lines have a slope of zero.

A vertical line has rise but zero run. Our denominator has a value of zero. We cannot divide by zero. Therefore, the slope of a vertical line is undefined.

1. Find the slope of a line that passes through points $\left(1,1\right)$ and $\left(5,9\right)$ .

We first plug each point into our formula.

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

$=\frac{9-1}{5-1}$

$=\frac{8}{4}$

$=2$

Therefore, our slope is 2.

2. Find the slope of a line that passes through $\left(1,-4\right)$ and $\left(2,-8\right)$ .

$m=\frac{\left({y}_{2}-{y}_{1}\right)}{\left({x}_{2}-{x}_{1}\right)}$

$=\frac{-8-\left(-4\right)}{2-1}$

$=\frac{-4}{1}$

$=-4$

3. If the slope of a line is 2 and it passes through $\left({x}_{1},4\right)$ and $\left(4,8\right)$ , calculate ${x}_{1}$ .

First, we'll insert each value into our formula.

$m=\frac{\left({y}_{2}-{y}_{1}\right)}{\left({x}_{2}-{x}_{1}\right)}$

$2=\frac{8-4}{4-{x}_{1}}$

$2=\frac{4}{4-{x}_{1}}$

Now we can multiply both sides by the $\left(4-{x}_{1}\right)$ piece to get:

$2\left(4-{x}_{1}\right)$

We distribute on the left side to get:

$8-2{x}_{1}=4$

Subtract both sides by 8:

$-2{x}_{1}=-4$

Divide both sides by -2:

${x}_{1}=2$

4. If a line passes through $\left(1,2\right)$ and $\left(1,5\right)$ , find the slope. What would this look like when graphed?

$m=\frac{\left({y}_{2}-{y}_{1}\right)}{\left({x}_{2}-{x}_{1}\right)}$

$=\frac{5-2}{1-1}$

$=\frac{3}{0}$

$\text{undefined}$

Since the slope is undefined, the graph would like a vertical line. Specifically, this is a vertical line that passes through 1 on the x-axis.

5. If a line passes through $\left(0,6\right)$ and $(5,6)$ , find the slope. What would this look like when graphed?

$m=\frac{\left({y}_{2}-{y}_{1}\right)}{\left({x}_{2}-{x}_{1}\right)}$

$=\frac{6-6}{5-0}$

$=\frac{0}{5}$

$=0$

Since the slope is zero, the graph will be a horizontal line passing through 6 on the y-axis.

College Algebra Diagnostic Tests

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