3.4 The Slope of a Line
Slopes of Lines
Definition A slope is a rate of change of variables. Those variables are typically x and y and the slope
of a line tells us how y changes with respect to x. In other words, it gives a vertical change with respect
to a horizontal change.
Visually, a slope describes the “slant” of a line.
Reading the graph from left to right, a positive slope means the line points upward, as if you are
walking up a mountain.
For a negative slope, the line points downward, as if you are walking down a mountain.
The greater the magnitude (so considering the value and not the sign), the steeper the slope.
Example A line with slope 5 is steeper (more slanted) than a line with slope 3 (since 5 > 3). Also, a
line with slope −5 is steeper than a line with slope −3 (since 5 > 3).
Finding the slope of a line
We use the letter m to describe the slope of a linear equation.
Formula For two points on the line, (x1 , y1 ) and (x2 , y2 ):
m=
∆y
vertical change
y2 − y1
=
=
∆x
horizontal change
x2 − x1
Note: ∆ is the uppercase greek letter which, mathematically, means change.
Example Find the slope of a line that goes through (3, −1) and (−6, 2).
Arbitrarily, let (x1 , y1 ) = (3, −1) and (x2 , y2 ) = (−6, 2).
Then m =
y2 − y1
2 − (−1)
3
1
=
=
=−
x2 − x1
−6 − 3
−9
3
Does it matter which point I choose for (x1 , y1 ) and (x2 , y2 )?
No (hence, the word arbitrarily above). To see this let (x1 , y1 ) = (−6, 2) and (x2 , y2 ) = (3, −1).
Then m =
y2 − y1
−1 − 2)
−3
1
=
=
=− .
x2 − x1
3 − (−6)
9
3
Notice what changed: the signs of the numerator and denominator.
Slopes of Horizontal Lines
Recall the form of a horizontal line: y = some number.
Consider the horizontal line y = 4. To determine the slope of the line, pick two points on the line and
plug into the slope formula.
Choosing (0, 4) and (1, 4):
m=
4−4
0
= = 0.
1−0
1
Notice that no matter what y-value the horizontal line is you will always get zero in the numerator of
the slope formula.
Therefore, the slope of a horizontal line is zero.
Slopes of Vertical Lines
Recall the form of a vertical line: x = some number.
Consider the vertical line x = 2. To determine the slope of the line, pick two points on the line and plug
into the slope formula.
Choosing (2, 0) and (2, 1):
m=
1
1−0
=
2−2
0
You cannot divide by zero and so the above is undefined.
Notice that no matter what x-value the vertical line is you will always get zero in the denominator of
the slope formula.
Therefore, the slope of a vertical line is undefined.
Standard form
Recall standard form for a linear equation: Ax + By = C.
Slope-intercept form
The slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line and b is the
y-intercept.
b is the y-intercept because when x = 0, y = b, corresponding to the point (0, b), which by definition is
the y-intercept.
Notice, that the name of this form is given by the information the form presents.
Example What is the slope and y-intercept of y = 5 − x?
Rewriting: y = −x + 5
By observation, m = −1 (the coefficient of x), and b = 5.
Therefore, the slope is −1 and the y-intercept is 5.
Switching forms
To go from slope-intercept form to standard form: put the variable terms on one side and the constant
on the other side.
Example Rewrite y = 2x + 5 in standard form.
y = 2x + 5
−2x + y = 5
To go from standard form to slope-intercept form: solve for y.
Example Rewrite 2x + y = 8 in slope-intercept form.
2x + y = 8
y = −2x + 8
Example Find the slope of 2x + 3y = 4.
Rewrite in slope-intercept form:
2x + 3y = 4
3y = 4 − 2x
y=
4 − 2x
3
Splitting into fractions: y =
4 2x
−
3
3
2
4
To see the slope clearly in y = mx + b form: y = − x +
3
3
2
The slope of the line is − .
3