<< Chapter < Page Chapter >> Page >

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.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 2. A line passes through the points (3, 0) and (3, 2).

What is the rise? The rise is 2. What is the run? The run is 0. What is the slope? m = rise run m = 2 0

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the slope of the line: x = −4 .

undefined

Got questions? Get instant answers now!

Find the slope of the line: y = 7 .

0

Got questions? Get instant answers now!

Quick guide to the slopes of lines

This figure shows four lines with arrows. The first line rises up and runs to the right. It has a positive slope. The second line falls down and runs to the right. It has a negative slope. The third line is neither rises nor falls, extending horizontally in either direction. It has a slope of zero. The fourth line is completely vertical, one end rising up and the other rising down, running neither to the left nor right. It has an undefined slope.

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 ( 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.

( x 1 , y 1 ) read ‘ x sub 1, y sub 1’ ( x 2 , y 2 ) read ‘ x sub 2, y sub 2’

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 ) .

The graph shows the x y coordinate plane. The x and y-axes run from 0 to 7. A line passes through the points (2, 3) and (7, 6), which are plotted and labeled. The ordered pair (2, 3) is labeled (x subscript 1, y subscript 1). The ordered pair (7, 6) is labeled (x subscript 2, y subscript 2). An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The first leg, from (2, 3) to (2, 6) is labeled y subscript 2 minus y subscript 1, 6 minus 3, and 3. The second leg, from (2, 3) to (7, 6), is labeled x subscript 2 minus x subscript 1, y minus 2, and 5.

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

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 = rise run . So m = 3 5 .

We rewrite the rise and run by putting in the coordinates m = 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 = 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 = y 2 y 1 x 2 x 1

We’ve shown that m = y 2 y 1 x 2 x 1 is really another version of m = rise 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
Srinu Reply
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.
Andrew Reply
divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
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!
Seera Reply
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
Juned Reply
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
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
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 Reply
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.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin Reply
Practice Key Terms 7

Get the best Elementary algebra course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask