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Find the slope of the line:

The graph shows the x y-coordinate plane. The x-axis runs from -8 to 1. The y-axis runs from -1 to 4. A line passes through the points “ordered pair -8,  1” and “ordered pair 0, 3”.

2 5

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Find the slope of the line:

The graph shows the x y-coordinate plane. The x-axis runs from -2 to 6. The y-axis runs from -2 to 4. A line passes through the points “ordered pair 4,  2” and “ordered pair 0, -1”.

3 4

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Find the slope from a graph.

  1. Locate two points on the line whose coordinates are integers.
  2. Starting with the point on the left, sketch a right triangle, 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. m = rise run

Find the slope of the line shown:

The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 4,  2” and “ordered pair 3, 3”.

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 angle to ( 3 , 3 ) as shown below.

The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 0,  5” and “ordered pair 3, 3”. Two line segments form a triangle with the line. A horizontal line connects “ordered pair 0, 3” and “ordered pair 3, 3 ”. A vertical line segment connects “ordered pair 0, 3” and “ordered pair 0, 5”. It is labeled “rise”.
Count the rise – it is negative. The rise is −2.
Count the run. The run is 3.
Use the slope formula. m = rise run
Substitute the values of the rise and run. m = −2 3
Simplify. m = 2 3
The slope of the line is 2 3 .

Notice that the slope is negative since the line slants downward from left to right.

What if we had chosen different points? Let’s find the slope of the line again, this time using different points. We will use the points ( −3 , 7 ) and ( 6 , 1 ) .
The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 0, 5” and  “ordered pair 3, 3”.  .

Starting at ( −3 , 7 ) , sketch a right triangle to ( 6 , 1 ) .

The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points “ordered pair 0, 5” and  “ordered pair 3, 3”. Two line segments form a triangle with the line. A vertical line connects “ordered pair 0, 3” and “ordered pair 3, 3 ”.  A vertical line segment connects “ordered pair 0, 3” and “ordered pair 0, 5”.
Count the rise. The rise is −6.
Count the run. The run is 9.
Use the slope formula. m = rise run
Substitute the values of the rise and run. m = −6 9
Simplify the fraction. m = 2 3
The slope of the line is 2 3 .

It does not matter which points you use—the slope of the line is always the same. The slope of a line is constant!

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Find the slope of the line:

The graph shows the x y-coordinate plane. The x-axis runs from -1 to 5. The y-axis runs from -6 to 1. A line passes through the points “ordered pair 3,  -6” and “ordered pair 0, -2”.

4 3

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Find the slope of the line:

The graph shows the x y-coordinate plane. The x-axis runs from -3 to 6. The y-axis runs from -3 to 2. A line passes through the points “ordered pair 5,  -2” and “ordered pair 0, 1”.

3 5

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The lines in the previous examples had y -intercepts with integer values, so it was convenient to use the y -intercept as one of the points we used to find the slope. In the next example, the y -intercept is a fraction. The calculations are easier if we use two points with integer coordinates.

Find the slope of the line shown: The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. A line passes through the points “ordered pair 2, 3” and “ordered pair 7, 6”.

Solution

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 angle to ( 7 , 6 ) as shown below.

The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. Two unlabeled points are drawn at  “ordered pair 2, 3” and  “ordered pair 7, 6”.  A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 2, 3” and “ordered pair 2, 6 ”.  It is labeled “rise”. A horizontal line segment connects “ordered pair 2, 6” and “ordered pair 7, 6”. It is labeled “run”.
Count the rise. The rise is 3.
Count the run. The run is 5.
Use the slope formula. m = rise run
Substitute the values of the rise and run. m = 3 5
The slope of the line is 3 5 .
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Find the slope of the line:

The graph shows the x y-coordinate plane. The x-axis runs from -4 to 2. The y-axis runs from -5 to 2. A line passes through the points “ordered pair -3, -4” and “ordered pair 1, 1”.

5 4

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Find the slope of the line:

The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -2 to 3. A line passes through the points “ordered pair 3, 2” and “ordered pair 1, -1”.

3 2

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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.

  • horizontal line y = b ; all the y -coordinates are the same.
  • vertical line x = a ; all the x -coordinates are the same.

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 in [link] . We’ll use the two points ( 0 , 4 ) and ( 3 , 4 ) to count the rise and run.

The graph shows the x y-coordinate plane. The x-axis runs from -1 to 5. The y-axis runs from -1 to 7. A horizontal line passes through the labeled points “ordered pair 0, 4” and “ordered pair 3, 4”.
What is the rise? The rise is 0.
What is the run? The run is 3.
What is the slope? m = rise run
m = 0 3
m = 0

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 .

Slope of a horizontal line

The slope of a horizontal line    , y = b , is 0 .

Now we’ll consider a vertical line, such as the line x = 3 , shown in [link] . We’ll use the two points ( 3 , 0 ) and ( 3 , 2 ) to count the rise and run.

Questions & Answers

how do they get the third part x = (32)5/4
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can someone help me with some logarithmic and exponential equations.
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ninjadapaul
20/(×-6^2)
Salomon
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it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
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I got X =-6
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ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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