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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 .


x = 8
This is a vertical line.
Its slope is undefined.

y = −5
This is a horizontal line.
It has slope 0.

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Find the slope of the line: x = −4 .


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Find the slope of the line: y = 7 .


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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

Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
Enith 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?
Alexandra Reply
I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons
Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Cecilia Reply
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year
Heidi Reply
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Elbert Reply
Avery and Caden have saved $27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
Leika Reply
1.2% of 27.000
i did 2.4%-7.2% i got 1.2%
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
I think Catherine is on the right track. Solve for x and y.
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Ericka Reply
Mario invested $475 in $45 and $25 stock shares. The number of $25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy?
Jawad Reply
let # of $25 shares be (x) and # of $45 shares be (y) we start with $25x + $45y=475, right? we are told the number of $25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
will every polynomial have finite number of multiples?
cricket Reply
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910.
David Reply
. A cashier has 54 bills, all of which are $10 or $20 bills. The total value of the money is $910. How many of each type of bill does the cashier have?
jojo Reply
whats the coefficient of 17x
Dwayne Reply
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
wow the exercise told me 17x solution is 14x lmao
thank you
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Mikaela Reply
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Sam Reply
Practice Key Terms 7

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