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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 .

Solution

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 .

undefined

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Find the slope of the line: y = 7 .

0

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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

In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
Jenise Reply
how do i set this up
Jenise
25%
Melissa
950,000-712500=237,500. 237,500/950,000 = .25 = 25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
Hi
Fernando
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
jamie Reply
6t+3
Melissa
6t +3
Bollywood
Tricia got a 6% raise on her weekly salary. The raise was $30 per week. What was her original salary?
Iris Reply
let us suppose her original salary is 'm'. so, according to the given condition, m*(6/100)=30 m= (30*100)/6 m= 500 hence, her original salary is $500.
Simply
28.50
Toi
thanks
Jeorge
How many pounds of nuts selling for $6 per pound and raisins selling for $3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound?
Valeria Reply
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that code $20 per square foot. How many square feet of each tile should she use so that the overal cost of he backsplash will be $10 per square foot?
Nia Reply
I need help with maths can someone help me plz.. is there a wats app group?
Cindy Reply
WY need
Fernando
How did you get $750?
Laura Reply
if y= 2x+sinx what is dy÷dx
formon25 Reply
does it teach you how to do algebra if you don't know how
Kate Reply
Liam borrowed a total of $35,000 to pay for college. He pays his parents 3% interest on the $8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total $35,000? (Round your answer to the nearest tenth of a percent.)
Lara Reply
exact definition of length by bilbao
francemichael Reply
the definition of length
francemichael Reply
literal meaning of length
francemichael
exact meaning of length
francemichael
exact meaning of length
francemichael
how many typos can we find...?
Chris Reply
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
Chris Reply
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
Practice Key Terms 7

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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