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This is a comparison and is therefore a ratio . Ratios can be expressed as fractions. Thus, a measure of the steepness of a line can be expressed as a ratio.

The slope of a line is defined as the ratio

Slope = change in y change in x

Mathematically, we can write these changes as

Slope = y 2 y 1 x 2 x 1

Finding the slope of a line

The slope of a nonvertical line passing through the points ( x 1 , y 1 ) and ( x 2 , y 2 ) is found by the formula
m = y 2 y 1 x 2 x 1

Sample set d

For the two given points, find the slope of the line that passes through them.

( 0 , 1 ) and ( 1 , 3 ) .

Looking left to right on the line we can choose ( x 1 , y 1 ) to be ( 0 , 1 ) , and ( x 2 , y 2 ) to be ( 1 , 3 ) . Then,

m = y 2 y 1 x 2 x 1 = 3 1 1 0 = 2 1 = 2

A graph of a line passing through two points with coordinates zero, one, and one, three with the upward change of two units and a horizontal change of one unit to the right.

This line has slope 2. It appears fairly steep. When the slope is written in fraction form, 2 = 2 1 , we can see, by recalling the slope formula, that as x changes 1 unit to the right (because of the + 1 ) y changes 2 units upward (because of the + 2 ).

m = change in y change in x = 2 1

Notice that as we look left to right, the line rises.

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( 2 , 2 ) and ( 4 , 3 ) .

Looking left to right on the line we can choose ( x 1 , y 1 ) to be ( 2 , 2 ) and ( x 2 , y 2 ) to be ( 4 , 3 ) . Then,

m = y 2 y 1 x 2 x 1 = 3 2 4 2 = 1 2

A graph of a line passing through two points with coordinates two, two, and four, three.

This line has slope 1 2 . Thus, as x changes 2 units to the right (because of the + 2 ), y changes 1 unit upward (because of the + 1 ).

m = change in y change in x = 1 2

Notice that in examples 1 and 2, both lines have positive slopes, + 2 and + 1 2 , and both lines rise as we look left to right.

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( 2 , 4 ) and ( 1 , 1 ) .

Looking left to right on the line we can choose ( x 1 , y 1 ) to be ( 2 , 4 ) and ( x 2 , y 2 ) to be ( 1 , 1 ) . Then,

m = y 2 y 1 x 2 x 1 = 1 4 1 ( 2 ) = 3 1 + 2 = 3 3 = 1

A graph of a line passing through two points with coordinates negative two, four, and one, one with a downward change of one unit and a horizontal change of one unit to the right.

This line has slope 1.

When the slope is written in fraction form, m = 1 = 1 + 1 , we can see that as x changes 1 unit to the right (because of the + 1 ), y changes 1 unit downward (because of the 1 ).
Notice also that this line has a negative slope and declines as we look left to right.

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( 1 , 3 ) and ( 5 , 3 ) .

m = y 2 y 1 x 2 x 1 = 3 3 5 1 = 0 4 = 0

A graph of a line parallel to x axis and passing through two points with coordinates one, three, and five, three.

This line has 0 slope. This means it has no rise and, therefore, is a horizontal line. This does not mean that the line has no slope, however.

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( 4 , 4 ) and ( 4 , 0 ) .

This problem shows why the slope formula is valid only for nonvertical lines.

m = y 2 y 1 x 2 x 1 = 0 4 4 4 = 4 0

A graph of a line parallel to y axis and passing through two points with coordinates four,zero and four, four.

Since division by 0 is undefined, we say that vertical lines have undefined slope. Since there is no real number to represent the slope of this line, we sometimes say that vertical lines have undefined slope , or no slope .

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Practice set d

Find the slope of the line passing through ( 2 , 1 ) and ( 6 , 3 ) . Graph this line on the graph of problem 2 below.

m = 3 1 6 2 = 2 4 = 1 2 .

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Find the slope of the line passing through ( 3 , 4 ) and ( 5 , 5 ) . Graph this line.

An xy-plane with gridlines

The line has slope 1 2 .

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Compare the lines of the following problems. Do the lines appear to cross? What is it called when lines do not meet (parallel or intersecting)? Compare their slopes. Make a statement about the condition of these lines and their slopes.

The lines appear to be parallel. Parallel lines have the same slope, and lines that have the same slope are parallel.

A graph of two parallel lines. One of the lines passes through two points with coordinates two, one and six, three. Another straight line passes through two points with coordinates three, four and five, five.

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Before trying some problems, let’s summarize what we have observed.

The equation y = m x + b is called the slope-intercept form of the equation of a line. The number m is the slope of the line and the point ( 0 , b ) is the y -intercept .

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The slope, m, of a line is defined as the steepness of the line, and it is the number of units that y changes when x changes 1 unit.

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The formula for finding the slope of a line through any two given points ( x 1 , y 1 ) and ( x 2 , y 2 ) is

m = y 2 - y 1 x 2 - x 1

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The fraction y 2 - y 1 x 2 - x 1 represents the Change in y Change in x .

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As we look at a graph from left to right, lines with positive slope rise and lines with negative slope decline.

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Parallel lines have the same slope.

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Horizontal lines have 0 slope.

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Vertical lines have undefined slope (or no slope).

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Exercises

For the following problems, determine the slope and y -intercept of the lines.

y = 3 x + 4

slope = 3 ;   y -intercept = ( 0 , 4 )

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y = 9 x + 1

slope = 9 ;   y -intercept = ( 0 , 1 )

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y = 4 x + 5

slope = 4 ;   y -intercept = ( 0 , 5 )

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y = 6 x 1

slope = 6 ;   y -intercept = ( 0 , 1 )

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y = x + 2

slope = 1 ;   y -intercept = ( 0 , 2 )

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4 y = 16 x + 20

slope = 4 ;   y -intercept = ( 0 , 5 )

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3 y = 12 x 27

slope = 4 ;   y -intercept = ( 0 , 9 )

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y = 2 7 x 12

slope = 2 7 ;   y -intercept = ( 0 , 12 )

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y = 4 5 x 4 7

slope = 4 5 ;   y -intercept = ( 0 , 4 7 )

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10 y = 12 x + 1

slope = 6 5 ;   y -intercept = ( 0 , 1 10 )

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y = x + 3

slope = 1;   y -intercept = ( 0 , 3 )

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5 x + 3 y = 6

slope = 5 3 ;   y -intercept = ( 0 , 2 )

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6 x 7 y = 12

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x + 4 y = 1

slope = 1 4 ;   y -intercept = ( 0 , 1 4 )

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For the following problems, find the slope of the line through the pairs of points.

( 1 , 6 ) , ( 4 , 9 )

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( 1 , 3 ) , ( 4 , 7 )

m = 4 3

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( 3 , 5 ) , ( 4 , 7 )

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( 6 , 1 ) , ( 2 , 8 )

m = 7 4

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( 0 , 5 ) , ( 2 , -6 )

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( -2 , 1 ) , ( 0 , 5 )

m = 2

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( 3 , -9 ) , ( 5 , 1 )

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( 4 , -6 ) , ( -2 , 1 )

m = 7 6

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( -5 , 4 ) , ( -1 , 0 )

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( -3 , 2 ) , ( -4 , 6 )

m = 4

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( 9 , 12 ) , ( 6 , 0 )

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( 0 , 0 ) , ( 6 , 6 )

m = 1

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( -2 , -6 ) , ( -4 , -1 )

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( -1 , -7 ) , ( -2 , -9 )

m = 2

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( -6 , -6 ) , ( -5 , -4 )

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( -1 , 0 ) , ( -2 , -2 )

m = 2

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( -4 , -2 ) , ( 0 , 0 )

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( 2 , 3 ) , ( 10 , 3 )

m = 0  ( horizontal line y = 3 )

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( 4 , -2 ) , ( 4 , 7 )

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( 8 , -1 ) , ( 8 , 3 )

No slope  ( vertical line at x = 8 )

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( 4 , 2 ) , ( 6 , 2 )

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( 5 , -6 ) , ( 9 , -6 )

m = 0  ( horizontal line at  y = 6 )

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Do lines with a positive slope rise or decline as we look left to right?

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Do lines with a negative slope rise or decline as we look left to right?

decline

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Make a statement about the slopes of parallel lines.

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Use a calculator. Calculator problems

For the following problems, determine the slope and y -intercept of the lines. Round to two decimal places.

3.8 x + 12.1 y = 4.26

slope = 0.31 y intercept = ( 0 , 0.35 )

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8.09 x + 5.57 y = 1.42

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10.813 x 17.0 y = 45.99

slope = 0.64 y intercept = ( 0 , 2.71 )

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6.003 x 92.388 y = 0.008

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For the following problems, find the slope of the line through the pairs of points. Round to two decimal places.

( 5.56 , 9.37 ) , ( 2.16 , 4.90 )

m = 1.31

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( 33.1 , 8.9 ) , ( 42.7 , 1.06 )

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( 155.89 , 227.61 ) , ( 157.04 , 227.61 )

m = 0  ( horizontal line at  y = 227.61 )

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( 0.00426 , 0.00404 ) , ( 0.00191 , 0.00404 )

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( 88.81 , 23.19 ) , ( 88.81 , 26.87 )

No slope  ( vertical line  x = 88.81 )

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( 0.0000567 , 0.0000567 ) , ( 0.00765 , 0.00764 )

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Exercises for review

( [link] ) Simplify ( x 2 y 3 w 4 ) 0 .

1 if x y w 0

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( [link] ) Solve the equation 3 x 4 ( 2 x ) 3 ( x 2 ) + 4 = 0 .

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( [link] ) When four times a number is divided by five, and that result is decreased by eight, the result is zero. What is the original number?

10

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( [link] ) Solve 3 y + 10 = x + 2 if x = 4 .

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( [link] ) Graph the linear equation x + y = 3 .
An xy coordinate plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates three, zero and zero, three.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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