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Slope

The number m is the coefficient of the variable x . The number m is called the slope of the line and it is the number of units that y changes when x is increased by 1 unit. Thus, if x changes by 1 unit, y changes by m units.
Since the equation y = m x + b contains both the slope of the line and the y -intercept , we call the form y = m x + b the slope-intercept form.

The slope-intercept form of the equation of a line

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

The slope is a measure of the steepness of a line

The word slope is really quite appropriate. It gives us a measure of the steepness of the line. Consider two lines, one with slope 1 2 and the other with slope 3. The line with slope 3 is steeper than is the line with slope 1 2 . Imagine your pencil being placed at any point on the lines. We make a 1-unit increase in the x -value by moving the pencil one unit to the right. To get back to one line we need only move vertically 1 2 unit, whereas to get back onto the other line we need to move vertically 3 units.

A graph of a line sloped up and to the right with lines illustrating an upward change of three units and a horizontal change of one unit to the right.

A graph of a line sloped up and to the right with lines illustrating an upward change of one half unit and a horizontal change of one unit to the right.

Sample set c

Find the slope and the y -intercept of the following lines.

y = 2 x + 7.

The line is in the slope-intercept form y = m x + b . The slope is m , the coefficient of x . Therefore, m = 2. The y -intercept is the point ( 0 , b ) . Since b = 7 , the y -intercept is ( 0 , 7 ) .

Slope : 2 y -intercept : ( 0 , 7 )

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y = 4 x + 1.

The line is in slope-intercept form y = m x + b . The slope is m , the coefficient of x . So, m = 4. The y -intercept is the point ( 0 , b ) . Since b = 1 , the y -intercept is ( 0 , 1 ).

Slope : 4 y -intercept : ( 0 , 1 )

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3 x + 2 y = 5.

The equation is written in general form. We can put the equation in slope-intercept form by solving for y .

3 x + 2 y = 5 2 y = 3 x + 5 y = 3 2 x + 5 2

Now the equation is in slope-intercept form.

Slope: 3 2 y -intercept: ( 0 , 5 2 )

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Practice set c

Find the slope and y -intercept of the line 2 x + 5 y = 15.

Solving for y we get y = 2 5 x + 3. Now, m = 2 5 and b = 3.

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The formula for the slope of a line

We have observed that the slope is a measure of the steepness of a line. We wish to develop a formula for measuring this steepness.

It seems reasonable to develop a slope formula that produces the following results:

Steepness of line 1 > steepness of line 2.

A graph of two lines sloped up and to the right in the first quadrant. Line with the lable 'Line one' has a steepness greater than the line with the lable 'Line two'.

Consider a line on which we select any two points. We’ll denote these points with the ordered pairs ( x 1 , y 1 ) and ( x 2 , y 2 ) . The subscripts help us to identify the points.

( x 1 , y 1 ) is the first point. Subscript 1 indicates the first point.
( x 2 , y 2 ) is the second point. Subscript 2 indicates the second point.

A graph of a line sloped up and to the right in the first quadrant passing through two points with coordinates x-one, y-one and x-two, y-two.

The difference in x values ( x 2 x 1 ) gives us the horizontal change, and the difference in y values ( y 2 y 1 ) gives us the vertical change. If the line is very steep, then when going from the first point to the second point, we would expect a large vertical change compared to the horizontal change. If the line is not very steep, then when going from the first point to the second point, we would expect a small vertical change compared to the horizontal change.

A graph of a line sloped up and to the right in a first quadrant. Lines illustrating an upward change of y-two minus y-one and a  horizontal change x-two minus x-one. Horzontal change is small as compared to  vertical change.

A graph of a line sloped up and to the right in a first quadrant. Lines illustrating an upward change of y-two minus y-one and a  horizontal change x-two minus x-one. Vertical change is small as compared to horzontal change

We are comparing changes. We see that we are comparing

The vertical change to the horizontal change The change in y to the change in x y 2 y 1 to x 2 x 1

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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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