# 4.4 Understand slope of a line  (Page 4/14)

 Page 4 / 14

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.

$\begin{array}{ccccc}\text{What is the rise?}\hfill & & \phantom{\rule{5em}{0ex}}& & \text{The rise is 2.}\hfill \\ \text{What is the run?}\hfill & & \phantom{\rule{5em}{0ex}}& & \text{The run is 0.}\hfill \\ \text{What is the slope?}\hfill & & \phantom{\rule{5em}{0ex}}& & \begin{array}{ccc}\hfill m& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \\ \hfill m& =\hfill & \frac{2}{0}\hfill \end{array}\hfill \end{array}$

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.

## Slope of a vertical line

The slope of a vertical line, $x=a$ , is undefined.

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.

Find the slope of the line: $x=-4.$

undefined

Find the slope of the line: $y=7.$

0

## Quick guide to the slopes of lines

Remember, we ‘read’ a line from left to right, just like we read written words in English.

## Use the slope formula to find the slope of a line between two points

Doing the Manipulative Mathematics activity “Slope of Lines Between Two Points” will help you develop a better understanding of how 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 $\left(x,y\right)$ gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol $\left(x,y\right)$ be used to represent two different points? Mathematicians use subscripts to distinguish the points.

$\begin{array}{ccc}\left({x}_{1},{y}_{1}\right)\hfill & & \text{read ‘}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sub 1,}\phantom{\rule{0.2em}{0ex}}y\phantom{\rule{0.2em}{0ex}}\text{sub 1’}\hfill \\ \left({x}_{2},{y}_{2}\right)\hfill & & \text{read ‘}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sub 2,}\phantom{\rule{0.2em}{0ex}}y\phantom{\rule{0.2em}{0ex}}\text{sub 2’}\hfill \end{array}$

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 $\left({x}_{1},{y}_{1}\right)$ to identify the first point and $\left({x}_{2},{y}_{2}\right)$ to identify the second point.

If we had more than two points, we could use $\left({x}_{3},{y}_{3}\right)$ , $\left({x}_{4},{y}_{4}\right)$ , 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 $\left(2,3\right)$ and $\left(7,6\right)$ .

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

On the graph, we counted the rise of 3 and the run of 5.

Notice that the rise of 3 can be found by subtracting the y -coordinates 6 and 3.

$3=6-3$

And the run of 5 can be found by subtracting the x -coordinates 7 and 2.

$5=7-2$

We know $m=\frac{\text{rise}}{\text{run}}$ . So $m=\frac{3}{5}$ .

We rewrite the rise and run by putting in the coordinates $m=\frac{6-3}{7-2}$ .

But 6 is ${y}_{2}$ , the y -coordinate of the second point and 3 is ${y}_{1}$ , the y -coordinate of the first point.

So we can rewrite the slope using subscript notation. $m=\frac{{y}_{2}-{y}_{1}}{7-2}$

Also, 7 is ${x}_{2}$ , the x -coordinate of the second point and 2 is ${x}_{1}$ , the x -coordinate of the first point.

So, again, we rewrite the slope using subscript notation. $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

We’ve shown that $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ is really another version of $m=\frac{\text{rise}}{\text{run}}$ . We can use this formula to find the slope of a line when we have two points on the line.

#### Questions & Answers

4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
divide 3x⁴-4x³-3x-1 by x-3
how to multiply the monomial
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Brandon has a cup of quarters and dimes with a total of 5.55\$. The number of quarters is five less than three times the number of dimes
app is wrong how can 350 be divisible by 3.
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
I'm getting "math processing error" on math problems. Anyone know why?
Can you all help me I don't get any of this
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Did anyone else have trouble getting in quiz link for linear inequalities?
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