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Understand Slope of a Line

Use geoboards to model slope

Use Geoboards to Model Slope

A geoboard is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.

We’ll start by stretching a rubber band between two pegs as shown in .

Doesn’t it look like a line?

Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle, as shown in

We carefully make a 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal.

To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown in .

If our geoboard and rubber band look just like the one shown in , the rise is 2. The rubber band goes up 2 units. (Each space is one unit.)

\[\begin{array}{lll}m & = & \frac{\text{rise}}{\text{run}} \\ m & = & \frac{2}{3}\end{array}\]

Example

Try it.

What is the slope of the line on the geoboard shown?

Solution

Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)

Start at the left peg and count the spaces up and to the right to reach the second peg.

The rise is 3.	\(m=\frac{3}{\text{run}}\)
The run is 4.	\(m=\frac{3}{4}\)
	The slope is \(\frac{3}{4}\).

This means that the line rises 3 units for every 4 units of run.

Example

Try it.

What is the slope of the line on the geoboard shown?

Solution

Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)

Start at the left peg and count the units down and to the right to reach the second peg.

The rise is −1.	\(=\frac{-1}{\text{run}}\)
The run is 3.	\(\begin{array}{l}m=\frac{-1}{3} \\ m=-\frac{1}{3}\end{array}\)
	The slope is \(-\frac{1}{3}\).

This means that the line drops 1 unit for every 3 units of run.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Use

Now, we’ll look at some graphs on the \(xy\)-coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards.

To find the slope, we must count out the rise and the run. But where do we start?

We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.

How to Use

Try it.

Find the slope of the line shown.

Solution

Example

Try it.

Find the slope of the line shown.

Solution

Locate two points on the graph whose coordinates are integers.	\((0,5)\) and \((3,3)\)
Which point is on the left?	\((0,5)\)
Starting at \((0,5)\), sketch a right triangle to \((3,3)\).
Count the rise—it is negative.	The rise is \(-2\).
Count the run.	The run is 3.
Use the slope formula.	\(m=\frac{\text{rise}}{\text{run}}\)
Substitute the values of the rise and run.	\(m=\frac{-2}{3}\)
Simplify.	\(m=-\frac{2}{3}\)
	The slope of the line is \(-\frac{2}{3}\).

So \(y\) decreases by 2 units as \(x\) increases by 3 units.

What if we used the points \((-3,7)\) and \((6,1)\) to find the slope of the line?

The rise would be \(-6\) and the run would be 9. Then \(m=\frac{-6}{9}\), and that simplifies to \(m=-\frac{2}{3}\). Remember, it does not matter which points you use—the slope of the line is always the same.

In the last two examples, the lines had y-intercepts with integer values, so it was convenient to use the y-intercept as one of the points to find the slope. In the next example, the y-intercept is a fraction. Instead of using that point, we’ll look for two other points whose coordinates are integers. This will make the slope calculations easier.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Find the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.

\[\begin{array}{llll}\text{Horizontal line}\ y=b & & & \text{Vertical line}\ x=a \\ \\ \text{y}\text{-coordinates are the same.} & & & \text{x}\text{-coordinates are the same.}\end{array}\]

So how do we find the slope of the horizontal line \(y=4\)? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens when we do this.

What is the rise?	The rise is \(0\).
Count the run.	The run is \(3\).
What is the slope?	\(\begin{array}{l}m=\frac{\text{rise}}{\text{run}} \\ m=\frac{0}{3} \\ m=0\end{array}\)
	The slope of the horizontal line \(y=4\) is \(0\).

All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.

The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away.

Now, we’ll consider a vertical line, the line.

What is the rise?	The rise is \(2\).
Count the run.	The run is \(0\).
What is the slope?	\(\begin{array}{l}m=\frac{\text{rise}}{\text{run}} \\ m=\frac{2}{0}\end{array}\)

But we can’t divide by 0. Division by 0 is not defined. So we say that the slope of the vertical line \(x=3\) is undefined.

The slope of any vertical line is undefined. When the x-coordinates of a line are all the same, the run is 0.

Example

Try it.

Find the slope of each line:

ⓐ \(x=8\) ⓑ \(y=-5\).

Solution

ⓐ \(x=8\)
This is a vertical line.
Its slope is undefined.
ⓑ \(y=-5\)
This is a horizontal line.
It has slope 0.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Use the Slope Formula to find the Slope of a Line Between Two Points

Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.

We have seen that an ordered pair \((x,y)\) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol \((x,y)\) be used to represent two different points? Mathematicians use subscripts to distinguish the points.

\[\begin{array}{lll}({x}_{1},{y}_{1}) & & \text{read ‘}\ x\ \text{sub 1,}\ y\ \text{sub 1’} \\ ({x}_{2},{y}_{2}) & & \text{read ‘}\ x\ \text{sub 2,}\ y\ \text{sub 2’}\end{array}\]

The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?

We will use \(({x}_{1},{y}_{1})\) to identify the first point and \(({x}_{2},{y}_{2})\) to identify the second point.

If we had more than two points, we could use \(({x}_{3},{y}_{3})\), \(({x}_{4},{y}_{4})\), and so on.

Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points \((2,3)\) and \((7,6)\).

Since we have two points, we will use subscript notation, \((\overset{{x}_{1},}{2,}\overset{{y}_{1}}{3})\)\((\overset{{x}_{2},{y}_{2}}{7,6})\).

\[3=6-3\]\[5=7-2\]

Example

Try it.

Use the slope formula to find the slope of the line between the points \((1,2)\) and \((4,5)\).

Solution

We'll call \((1,2)\) point #1 and \((4,5)\) point #2.	\((\overset{{x}_{1},{y}_{1}}{1,2})\ {(\overset{{x}_{2},{y}_{2}}{4,5})}_{}\)
Use the slope formula.	\(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\).
Substitute the values.
\(y\) of the second point minus \(y\) of the first point	\(m=\frac{5-2}{{x}_{2}-{x}_{1}}\).
\(x\) of the second point minus \(x\) of the first point	\(m=\frac{5-2}{4-1}\).
Simplify the numerator and the denominator.	\(m=\frac{3}{3}\).
Simplify.	\(m=1\).

Let’s confirm this by counting out the slope on a graph using \(m=\frac{\text{rise}}{\text{run}}\).

It doesn’t matter which point you call point #1 and which one you call point #2. The slope will be the same. Try the calculation yourself.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Graph a Line Given a Point and the Slope

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

One other method we can use to graph lines is called the point–slope method. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.

How To Graph a Line Given a Point and The Slope

Try it.

Graph the line passing through the point \((1,-1)\) whose slope is \(m=\frac{3}{4}\).

Solution

Example

Try it.

Graph the line with y-intercept 2 whose slope is \(m=-\frac{2}{3}\).

Solution

Plot the given point, the y-intercept, \((0,2)\).

Identify the rise and the run.	\(m=-\frac{2}{3}\)
	\(\frac{\text{rise}}{\text{run}}=\frac{-2}{3}\)
	\(\text{rise}=-2\)
	\(\text{run}=3\)

Count the rise and the run. Mark the second point.

Connect the two points with a line.

You can check your work by finding a third point. Since the slope is \(m=-\frac{2}{3}\), it can be written as \(m=\frac{2}{-3}\). Go back to \((0,2)\) and count out the rise, 2, and the run, \(-3\).

Example

Try it.

Graph the line passing through the point \((-1,-3)\) whose slope is \(m=4.\)

Solution

Plot the given point.

Identify the rise and the run.	\(m=4\)
Write 4 as a fraction.	\(\frac{\text{rise}}{\text{run}}=\frac{4}{1}\)
	\(\text{rise}=4,\text{run}=1\)

Count the rise and run and mark the second point.

Connect the two points with a line.

You can check your work by finding a third point. Since the slope is \(m=4\), it can be written as \(m=\frac{-4}{-1}\). Go back to \((-1,-3)\) and count out the rise, \(-4\), and the run, \(-1\).

Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.

Example

Try it.

The ‘pitch’ of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?

Solution

Use the slope formula.	\(m=\frac{\text{rise}}{\text{run}}\)
Substitute the values for rise and run.	\(m=\frac{9}{18}\)
Simplify.	\(m=\frac{1}{2}\)
The slope of the roof is \(\frac{1}{2}\).
	The roof rises 1 foot for every 2 feet of horizontal run.

Example

Try it.

Have you ever thought about the sewage pipes going from your house to the street? They must slope down \(\frac{1}{4}\) inch per foot in order to drain properly. What is the required slope?

Solution

Use the slope formula.	\(\begin{array}{l}m=\frac{\text{rise}}{\text{run}} \\ m=\frac{-\frac{1}{4}\text{inch}}{\text{1 foot}} \\ m=\frac{-\frac{1}{4}\text{inch}}{\text{12 inches}}\end{array}\)
Simplify.	\(m=-\frac{1}{48}\)
	The slope of the pipe is \(-\frac{1}{48}\).

The pipe drops 1 inch for every 48 inches of horizontal run.

Key Concepts

Find the Slope of a Line from its Graph using \(m=\frac{\text{rise}}{\text{run}}\)
1. Locate two points on the line whose coordinates are integers.
2. Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point.
3. Count the rise and the run on the legs of the triangle.
4. Take the ratio of rise to run to find the slope.
Graph a Line Given a Point and the Slope
1. Plot the given point.
2. Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.
3. Starting at the given point, count out the rise and run to mark the second point.
4. Connect the points with a line.
Slope of a Horizontal Line
- The slope of a horizontal line, \(y=b\), is 0.
Slope of a vertical line
- The slope of a vertical line, \(x=a\), is undefined

Understand Slope of a Line

Use Geoboards to Model Slope

In the following exercises, find the slope modeled on each geoboard.

Try it.

Solution

\(\frac{1}{4}\)

Try it.

Solution

\(\frac{2}{3}\)

Try it.

Solution

\(\frac{-3}{2}=-\frac{3}{2}\)

Try it.

Solution

\(-\frac{3}{4}\)

Try it.

In the following exercises, model each slope. Draw a picture to show your results.

Try it.

\(\frac{2}{3}\)

Solution

Try it.

\(\frac{3}{4}\)

Try it.

\(\frac{1}{4}\)

Solution

Try it.

\(\frac{4}{3}\)

Try it.

\(-\frac{1}{2}\)

Solution

Try it.

\(-\frac{3}{4}\)

Try it.

\(-\frac{2}{3}\)

Solution

Try it.

\(-\frac{3}{2}\)

Use \(m=\frac{\text{rise}}{\text{run}}\) to find the Slope of a Line from its Graph

In the following exercises, find the slope of each line shown.

Try it.

Solution

\(\frac{2}{5}\)

Try it.

Solution

\(\frac{5}{4}\)

Try it.

Solution

\(-\frac{1}{3}\)

Try it.

Solution

\(-\frac{3}{4}\)

Try it.

Solution

\(\frac{3}{4}\)

Try it.

Solution

\(-3\)

Try it.

Solution

\(-\frac{2}{3}\)

Try it.

Solution

\(\frac{1}{4}\)

Try it.

Find the Slope of Horizontal and Vertical Lines

In the following exercises, find the slope of each line.

Try it.

\(y=3\)

Solution

Try it.

\(y=1\)

Try it.

\(x=4\)

Solution

undefined

Try it.

\(x=2\)

Try it.

\(y=-2\)

Solution

Try it.

\(y=-3\)

Try it.

\(x=-5\)

Solution

undefined

Try it.

\(x=-4\)

Use the Slope Formula to find the Slope of a Line between Two Points

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

Try it.

\((1,4),(3,9)\)

Solution

\(\frac{5}{2}\)

Try it.

\((2,3),(5,7)\)

Try it.

\((0,3),(4,6)\)

Solution

\(\frac{3}{4}\)

Try it.

\((0,1),(5,4)\)

Try it.

\((2,5),(4,0)\)

Solution

\(-\frac{5}{2}\)

Try it.

\((3,6),(8,0)\)

Try it.

\((-3,3),(4,-5)\)

Solution

\(-\frac{8}{7}\)

Try it.

\((-2,4),(3,-1)\)

Try it.

\((-1,-2),(2,5)\)

Solution

\(\frac{7}{3}\)

Try it.

\((-2,-1),(6,5)\)

Try it.

\((4,-5),(1,-2)\)

Solution

\(-1\)

Try it.

\((3,-6),(2,-2)\)

Graph a Line Given a Point and the Slope

In the following exercises, graph each line with the given point and slope.

Try it.

\((1,-2)\); \(m=\frac{3}{4}\)

Solution

Try it.

\((1,-1)\); \(m=\frac{2}{3}\)

Try it.

\((2,5)\); \(m=-\frac{1}{3}\)

Solution

Try it.

\((1,4)\); \(m=-\frac{1}{2}\)

Try it.

\((-3,4)\); \(m=-\frac{3}{2}\)

Solution

Try it.

\((-2,5)\); \(m=-\frac{5}{4}\)

Try it.

\((-1,-4)\); \(m=\frac{4}{3}\)

Solution

Try it.

\((-3,-5)\); \(m=\frac{3}{2}\)

Try it.

y-intercept 3; \(m=-\frac{2}{5}\)

Solution

Try it.

y-intercept 5; \(m=-\frac{4}{3}\)

Try it.

x-intercept \(-2\); \(m=\frac{3}{4}\)

Solution

Try it.

x-intercept \(-1\); \(m=\frac{1}{5}\)

Try it.

\((-3,3)\); \(m=2\)

Solution

Try it.

\((-4,2)\); \(m=4\)

Try it.

\((1,5)\); \(m=-3\)

Solution

Try it.

\((2,3)\); \(m=-1\)

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: \(\frac{1-4}{8-2}\).
If you missed this problem, review .

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\(-\frac{1}{2}\)
Divide: \(\frac{0}{4},\frac{4}{0}\).
If you missed this problem, review .

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0, undefined
Simplify: \(\frac{15}{-3},\frac{-15}{3},\frac{-15}{-3}\).
If you missed this problem, review .

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\(-5,-5,5\)
What is the slope of the line on the geoboard shown?

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Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)
Start at the left peg and count the spaces up and to the right to reach the second peg.

The rise is 3. \(m=\frac{3}{\text{run}}\)
The run is 4. \(m=\frac{3}{4}\)
The slope is \(\frac{3}{4}\).
This means that the line rises 3 units for every 4 units of run.
What is the slope of the line on the geoboard shown?

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\(\frac{4}{3}\)
What is the slope of the line on the geoboard shown?

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\(\frac{1}{4}\)
What is the slope of the line on the geoboard shown?

Odhaliť odpoveď

Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)
Start at the left peg and count the units down and to the right to reach the second peg.

The rise is −1. \(=\frac{-1}{\text{run}}\)
The run is 3. \(\begin{array}{l}m=\frac{-1}{3} \\ m=-\frac{1}{3}\end{array}\)
The slope is \(-\frac{1}{3}\).
This means that the line drops 1 unit for every 3 units of run.
What is the slope of the line on the geoboard?

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\(-\frac{2}{3}\)
What is the slope of the line on the geoboard?

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\(-\frac{4}{3}\)
Use a geoboard to model a line with slope \(\frac{1}{2}\).

Odhaliť odpoveď

To model a line on a geoboard, we need the rise and the run.
Use the slope formula. \(m=\frac{\text{rise}}{\text{run}}\)
Replace \(m\) with \(\frac{1}{2}\). \(\frac{1}{2}=\frac{\text{rise}}{\text{run}}\)
So, the rise is 1 and the run is 2.
Start at a peg in the lower left of the geoboard.
Stretch the rubber band up 1 unit, and then right 2 units.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is \(\frac{1}{2}\).
Model the slope \(m=\frac{1}{3}\). Draw a picture to show your results.

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Model the slope \(m=\frac{3}{2}\). Draw a picture to show your results.

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Use a geoboard to model a line with slope \(\frac{-1}{4}.\)

Odhaliť odpoveď

Use the slope formula. \(m=\frac{\text{rise}}{\text{run}}\)
Replace \(m\) with \(\frac{-1}{4}\). \(\frac{-1}{4}=\frac{\text{rise}}{\text{run}}\)
So, the rise is \(-1\) and the run is 4.
Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down.
We stretch the rubber band down 1 unit, then go to the right 4 units, as shown.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is \(\frac{-1}{4}\).
Model the slope \(m=\frac{-2}{3}\). Draw a picture to show your results.

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Model the slope \(m=\frac{-1}{3}\). Draw a picture to show your results.

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Find the slope of the line shown.
Find the slope of the line shown.

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\(\frac{2}{5}\)
Find the slope of the line shown.

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\(\frac{3}{4}\)

Find the slope of the line shown.

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Locate two points on the graph whose coordinates are integers.	\((0,5)\) and \((3,3)\)
Which point is on the left?	\((0,5)\)
Starting at \((0,5)\), sketch a right triangle to \((3,3)\).
Count the rise—it is negative.	The rise is \(-2\).
Count the run.	The run is 3.
Use the slope formula.	\(m=\frac{\text{rise}}{\text{run}}\)
Substitute the values of the rise and run.	\(m=\frac{-2}{3}\)
Simplify.	\(m=-\frac{2}{3}\)
	The slope of the line is \(-\frac{2}{3}\).

So \(y\) decreases by 2 units as \(x\) increases by 3 units.

What if we used the points \((-3,7)\) and \((6,1)\) to find the slope of the line?

Find the slope of the line shown.

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\(-\frac{4}{3}\)
Find the slope of the line shown.

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\(-\frac{3}{5}\)

Find the slope of the line shown.

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Locate two points on the graph whose coordinates are integers.	\((2,3)\) and \((7,6)\)
Which point is on the left?	\((2,3)\)
Starting at \((2,3)\), sketch a right triangle to \((7,6)\).
Count the rise.	The rise is 3.
Count the run.	The run is 5.
Use the slope formula.	\(m=\frac{\text{rise}}{\text{run}}\)
Substitute the values of the rise and run.	\(m=\frac{3}{5}\)
	The slope of the line is \(\frac{3}{5}\).

This means that \(y\) increases 5 units as \(x\) increases 3 units.

When we used geoboards to introduce the concept of slope, we said that we would always start with the point on the left and count the rise and the run to get to the point on the right. That way the run was always positive and the rise determined whether the slope was positive or negative.

What would happen if we started with the point on the right?

Let’s use the points \((2,3)\) and \((7,6)\) again, but now we’ll start at \((7,6)\).

Count the rise.	The rise is \(-3\).
Count the run. It goes from right to left, so it is negative.	The run is \(-5\).
Use the slope formula.	\(m=\frac{\text{rise}}{\text{run}}\)
Substitute the values of the rise and run.	\(m=\frac{-3}{-5}\)
	The slope of the line is \(\frac{-3}{-5}\).

It does not matter where you start—the slope of the line is always the same.

Find the slope of the line shown.

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\(\frac{5}{4}\)
Find the slope of the line shown.

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\(\frac{3}{2}\)
Find the slope of each line:
ⓐ \(x=8\) ⓑ \(y=-5\).
Odhaliť odpoveď
1. ⓐ \(x=8\)
  This is a vertical line.
  Its slope is undefined.
2. ⓑ \(y=-5\)
  This is a horizontal line.
  It has slope 0.
Find the slope of the line: \(x=-4.\)

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undefined
Find the slope of the line: \(y=7.\)

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0

Use the slope formula to find the slope of the line between the points \((1,2)\) and \((4,5)\).

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We'll call \((1,2)\) point #1 and \((4,5)\) point #2.	\((\overset{{x}_{1},{y}_{1}}{1,2})\ {(\overset{{x}_{2},{y}_{2}}{4,5})}_{}\)
Use the slope formula.	\(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\).
Substitute the values.
\(y\) of the second point minus \(y\) of the first point	\(m=\frac{5-2}{{x}_{2}-{x}_{1}}\).
\(x\) of the second point minus \(x\) of the first point	\(m=\frac{5-2}{4-1}\).
Simplify the numerator and the denominator.	\(m=\frac{3}{3}\).
Simplify.	\(m=1\).

Let’s confirm this by counting out the slope on a graph using \(m=\frac{\text{rise}}{\text{run}}\).

It doesn’t matter which point you call point #1 and which one you call point #2. The slope will be the same. Try the calculation yourself.

Use the slope formula to find the slope of the line through the points: \((8,5)\) and \((6,3)\).

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1
Use the slope formula to find the slope of the line through the points: \((1,5)\) and \((5,9)\).

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1

Use the slope formula to find the slope of the line through the points \((-2,-3)\) and \((-7,4)\).

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We'll call \((-2,-3)\) point #1 and \((-7,4)\) point #2.	\((\overset{{x}_{1},{y}_{1}}{-2,-3})\ {(\overset{{x}_{2},{y}_{2}}{-7,4})}_{}\)
Use the slope formula.	\(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\).
Substitute the values.
\(y\) of the second point minus \(y\) of the first point	\(m=\frac{4-(-3)}{{x}_{2}-{x}_{1}}\).
\(x\) of the second point minus \(x\) of the first point	\(m=\frac{4-(-3)}{-7-(-2)}\).
Simplify.	\(\begin{array}{l}m=\frac{7}{-5} \\ m=-\frac{7}{5}\end{array}\)

Let’s verify this slope on the graph shown.

\[\begin{array}{lll}m & = & \frac{\text{rise}}{\text{run}} \\ m & = & \frac{-7}{5} \\ m & = & -\frac{7}{5}\end{array}\]

Use the slope formula to find the slope of the line through the points: \((-3,4)\) and \((2,-1).\)

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\(-1\)
Use the slope formula to find the slope of the line through the pair of points: \((-2,6)\) and \((-3,-4)\).

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10
Graph the line passing through the point \((1,-1)\) whose slope is \(m=\frac{3}{4}\).
Graph the line passing through the point \((2,-2)\) with the slope \(m=\frac{4}{3}\).

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Graph the line passing through the point \((-2,3)\) with the slope \(m=\frac{1}{4}\).

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Graph the line with y-intercept 2 whose slope is \(m=-\frac{2}{3}\).

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Plot the given point, the y-intercept, \((0,2)\).
Identify the rise and the run. \(m=-\frac{2}{3}\)
\(\frac{\text{rise}}{\text{run}}=\frac{-2}{3}\)
\(\text{rise}=-2\)
\(\text{run}=3\)
Count the rise and the run. Mark the second point.
Connect the two points with a line.
You can check your work by finding a third point. Since the slope is \(m=-\frac{2}{3}\), it can be written as \(m=\frac{2}{-3}\). Go back to \((0,2)\) and count out the rise, 2, and the run, \(-3\).
Graph the line with the y-intercept 4 and slope \(m=-\frac{5}{2}.\)

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Graph the line with the x-intercept \(-3\) and slope \(m=-\frac{3}{4}\).

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Graph the line passing through the point \((-1,-3)\) whose slope is \(m=4.\)

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Plot the given point.
Identify the rise and the run. \(m=4\)
Write 4 as a fraction. \(\frac{\text{rise}}{\text{run}}=\frac{4}{1}\)
\(\text{rise}=4,\text{run}=1\)
Count the rise and run and mark the second point.
Connect the two points with a line.
You can check your work by finding a third point. Since the slope is \(m=4\), it can be written as \(m=\frac{-4}{-1}\). Go back to \((-1,-3)\) and count out the rise, \(-4\), and the run, \(-1\).

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Understand Slope of a Line

Use geoboards to model slope
Use
Find the slope of horizontal and vertical lines
Use the slope formula to find the slope of a line between two points
Graph a line given a point and the slope
Solve slope applications
Locate two points on the line whose coordinates are integers.
Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

Use the slope formula.	\(m=\frac{\text{rise}}{\text{run}}\)
Replace \(m\) with \(\frac{1}{2}\).	\(\frac{1}{2}=\frac{\text{rise}}{\text{run}}\)

Use the slope formula.	\(m=\frac{\text{rise}}{\text{run}}\)
Replace \(m\) with \(\frac{-1}{4}\).	\(\frac{-1}{4}=\frac{\text{rise}}{\text{run}}\)