34 THE SLOPE OF A

Slope  =  _Vertical leg_ Horizontal leg 
=  Δy Δx 
= 
For example,
If the value of y changes by 2 units when the value of x changes by 3,
then the slope of that line is  2 3 
. 
What does slope  2 3 
mean? It indicates the rate at which a change 
in the value of x produces a change in the value of y. 2 units of y per  for every  3 units of x.
For every 3 units that line moves to the right, it will move up 2. That will be true between any two points on that line. Over 6 and up 4, over 15 and up 10. Because a straight line has one and only one slope. (Theorem 8.1 of Precalculus.)
In each line above, the xcoördinate has increased by 1 unit. In the line on the left, however, the value of y has increased much more than in the line on the right. The line on the left has a greater slope than the line on the right. The value of y has changed at a much greater rate.
If the xaxis represents time and the yaxis distance, as is the case in many applications, then the rate of change of y with respect to x  of distance with respect to time  is called speed or velocity. So many miles per hour, or meters per second.
Up or down?
Which line do we say is sloping "up"? And which is sloping "down"?
Since we imagine moving along the xaxis from left to right, we say that the line on the left is sloping up, and the line on the right, down.
What is more, a line that slopes up has a positive slope. While a line that slopes down has a negative slope.
For, both the x and ycoördinates of B are greater than the coördinates of A, so that both Δx and Δy are positive. Therefore their quotient, which is the slope, is positive.
But while the xcoördinate of D is greater than the xcoördinate of C, so that Δx is positive, the ycoördinate of D is less than the ycoördinate of C, so that Δy, 5 − 8, is negative. Therefore that quotient is negative.
Problem 1. What number is the slope of each line?
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a)  b)  
3 4 
4 3 
c)  d)  
−  4 3 
−  3 4 
Horizontal and vertical lines
What number is the slope of a horizontal line  that is, a line parallel to the xaxis? And what is the slope of a vertical line?
A horizontal line has slope 0, because even though the value of x changes, the value of y does not. Δy = 0.
Δy Δx 
=  0 Δx 
=  0.  (Lesson 5) 
For a vertical line, however, the slope is not defined. The slope tells how the ycoördinate changes when the xcoördinate changes. But the xcoördinate does not change  Δx = 0. A vertical line does not have a slope.
Δy Δx 
=  Δy 0 
=  No value.  (Lesson 5) 
Problem 2.
a) Which numbered lines have a positive slope? 2 and 4.
b) Which numbered lines have a negative slope? 1 and 3.
c) What slope has the horizontal line 5? 0.
c) What slope has the vertical line 6? It does not have a slope.
Example 1. Calculate the slope of the line that passes through the points (3, 6) and (1, 2)
Solution. To do this problem, here again is the definition of the slope. It is the number
Δy Δx 
=  Difference of ycoördinates Difference of xcoördinates 
= 
Therefore, the slope of the line passing through (3, 6) and (1, 2) is:
Δy Δx 
=  6 − 2 3 − 1 
=  4 2 
=  2 1 
= 2. 
Note: It does not matter which point we call the first and which the second. But if we calculate Δy starting with (3, 6), then we must calculate Δx also starting with (3, 6).
As for the meaning of slope 2: On the straight line that joins those two points, for every 1 unit the value of x changes, the value of y will change by 2 units. That is the rate of change of y with respect to x. 2 for every 1.
Problem 3. Calculate the slope of the line that joins these points.
a)  (1, 5) and (4, 17)  b)  (−3, 10) and (−2, 7)  
17 − 5 4 − 1 
=  12 3 
=  4  _ 10 − 7 _ −3 − (−2) 
=  3 −1 
=  −3  
c)  (1, −1) and (−7, −5)  d)  (2, −9) and (−2, −5)  
−5 − (−1) −7 − 1 
=  −4 −8 
=  ½  −9 − (−5) 2 − (−2) 
=  −4 4 
=  −1 
The slopeintercept form
This linear form
y = ax + b
is called the slopeintercept form of the equation of a straight line. Because, as we can prove (Topic 9 of Precalculus): a is the slope of the line, and b is the yintercept.
Problem 4. What number is the slope of each line, and what is the meaning of each slope?
a) y = 5x − 2
The slope is 5. This means that y increases 5 units for every 1 unit x increases. That is the rate of change of y with respect to x.
b) y = −  2 3 
x + 4 
The slope is −  2 3 
. This means that y decreses 2 units for every 
3 units of x..
Problem 5.
a) Write the equation of the straight line whose slope is 3 and whose
a) yintercept is 1.
y = 3x + 1
b) Write the equation of the straight line whose slope is −1 and whose
a) yintercept is −2.
y = −x − 2
c)  Write the equation of the straight line whose slope is  2 3 
and which 
passes through the origin. 
y =  2 3 
x. The yintercept b is 0. 
Problem 6. Sketch the graph of y = −2x.
This is a straight line of slope −2  over 1 and down 2  that passes through the origin: b = 0.
(Compare Lesson 33, Problem 15.)
The general form
This linear form
Ax + By + C = 0
where A, B, C are integers (Lesson 2), is called the general form of the equation of a straight line.
Problem 7. What number is the slope of each line, and what is the meaning of each slope?
a) x + y − 5 = 0
This line is in the general form. It is only when the line is in the slopeintercept form, y = ax + b, that the slope is a. Therefore, on solving for y: y = −x + 5. The slope therefore is −1. This means that the value of y decreases 1 unit for every unit that the value of x increases.
b)  2x − 3y + 6  =  0  
−3y  =  −2x − 6  
y  =  2 3 
x + 2,  on dividing every term by −3. 
The slope therefore is  2 3 
. This means that the line goes up 2 
for every 3 units it goes over.
c)  Ax + By + C  =  0  
By  =  −Ax − C  
y  =  −  A B 
x  −  C B 
, on dividing every term by B. 
The slope is −  A B 
. 
We can view that as a formula for the slope when the equation is in the general form. For example, if the equation is
4x − 5y + 2 = 0,
then the slope is
−  4 −5 
=  4 5 
. 
Parallel and perpendicular lines
Straight lines will be parallel if they have the same slope. The following are equations of parallel lines:
y = 3x + 1 and y = 3x − 8.
They have the same slope 3.
Straight lines will be perpendicular if
1)  their slopes have opposite signs  one positive and one negative, and  
2)  they are reciprocals of one another. 
That is:
If m is the slope of one line, then a perpendicular line has slope −  1 m 
. 
To be specific, if a line has slope 4, then every line that is perpendicular to it has slope −¼.
(We will prove that below.)
Problem 8. Which of these lines are parallel and which are perpendicular?
a) y = 2x + 3  b) y = −2x + 3  c) y = ½x + 3  d) y = 2x − 3 
a) and d) are parallel. b) and c) are perpendicular.
Problem 9. If a line has slope 5, then what is the slope of a line that is
perpendicular to it?  − 
1 5 
Problem 10. If a line has slope −  2 3 
, then what is the slope of a  
perpendicular line? 
3 2 
Problem 11. If a line has equation y = 6x − 5, then what is the slope  
of a perpendicular line?  − 
1 6 
Theorem: The slopes of perpendicular lines
If two straight lines are perpendicular to one another, then the product of their slopes is −1.
That is: If the slope of one line is m, then the slope of the perpendicular line
is −  1 m 
. 
Let L_{1} be a straight line, and let the perpendicular straight line L_{2} cross L_{1} at the point A.
Let L_{1} have slope m_{1}, and let L_{2} have slope m_{2}. Assume that m_{1} is positive. Then m_{2}, as we will see, must be negative.
Draw a straight line AB of length 1 parallel to the xaxis, and draw BC at right angles to AB equal in length to m_{1}.
Extend CB in a straight line to join L_{2} at D.
Now, since the straight line L_{2} has one slope m_{2} (Theorem 8.1 of Precalculus), the length of BD will be m_{2}. For in going from A to D on L_{2}, we go over 1 and down m_{2}.
That is, m_{2} is a negative number.
(Same figure.)
Angle CAD is a right angle. Therefore angle a is the complement of angle ß.
But triangle ABD is rightangled, and therefore the angle at D is also the complement of angle ß;
therefore the angle at D is equal to angle a.
The right triangles ABC, ABD therefore are similar (Topic 5 of Trigonometry),
and the sides opposite the equal angles are proportional:
This implies
m_{1}m_{2} = 1.
But m_{2} is negative. Therefore,
m_{1}m_{2} = −1.
Which is what we wanted to prove.
Next Lesson: Simultaneous linear equations
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