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The Slope of a Line
Mathematicians have developed a useful measure of the steepness of a line,
called the slope of the line. Slope compares the vertical change (the rise) to
the horizontal change (the run) when moving from one fixed point to
another along the line. A ratio comparing the change in y (the rise) with the
change in x (the run) is used calculate the slope of a line.
Definition of Slope
The slope of the line through the distinct points (x1
, y1
) and (x2
, y2
) is
where x1
– x2
= 0.
Definition of Slope
The slope of the line through the distinct points (x1
, y1
) and (x2
, y2
) is
where x1
– x2
= 0.
Change in y
Change in x
=
Rise
Run
=
y2 – y1
x2 – x1
(x1, y1)
x1
y1
x2
y2
(x2, y2)Rise
y2 – y1
Run
x2 – x1
x
y
Chapter 1. Graphs, Functions, & Models

Sample Problems:
• Find the slope of the line thru the points
given:
(-3,-1) and (-2,4)
(-3,4) and (2,-2)
m=
4− −1( )( )
−2− −3( )( )
=
5
1
=5
m=
−2−4( )
2− −3( )( )
=
−6
5

The Possibilities for a Line’s Slope
Positive Slope
x
y
m > 0
Line rises from left to right.
Zero Slope
x
y
m = 0
Line is horizontal.
m is
undefined
Undefined Slope
x
y
Line is vertical.
Negative Slope
x
y
m < 0
Line falls from left to right.

Point-Slope Form of the Equation of a Line
The point-slope equation of a non-vertical line of slope m that passes
through the point (x1
, y1
) is
y – y1
= m(x – x1
).
The point-slope equation of a non-vertical line of slope m that passes
through the point (x1
, y1
) is
y – y1
= m(x – x1
).

Example: Writing the Point-Slope Equation of a Line
Write the point-slope form of the equation of the line passing through (-1,3)
with a slope of 4. Then solve the equation for y.
Solution We use the point-slope equation of a line with m = 4, x1
= -1, and
y1
= 3.
This is the point-slope form of the equation.y – y1 = m(x – x1)
Substitute the given values. Simply.y – 3 = 4[x – (-1)]
We now have the point-slope form of the equation for the
given line.
y – 3 = 4(x + 1)
We can solve the equation for y by applying the distributive property.
y – 3 = 4x + 4
y = 4x + 7 Add 3 to both sides.

Slope-Intercept Form of the Equation of a
LineThe slope-intercept equation of a non-vertical line with slope m and y-
intercept b is
y = mx + b.
The slope-intercept equation of a non-vertical line with slope m and y-
intercept b is
y = mx + b.

Equations of Horizontal and Vertical Lines
Equation of a Horizontal Line
A horizontal line is given by an equation of the form
y = b
where b is the y-intercept. Note: m = 0.
Equation of a Horizontal Line
A horizontal line is given by an equation of the form
y = b
where b is the y-intercept. Note: m = 0.
Equation of a Vertical Line
A vertical line is given by an equation of the form
x = a
where a is the x-intercept. Note: m is undefined.
Equation of a Vertical Line
A vertical line is given by an equation of the form
x = a
where a is the x-intercept. Note: m is undefined.

General Form of the Equation of the a Line
Every line has an equation that can be written in the general form
Ax + By + C = 0
Where A, B, and C are three integers, and A and B are not both zero. A
must be positive.
Every line has an equation that can be written in the general form
Ax + By + C = 0
must be positive.
Every line has an equation that can be written in the standard form
Ax + By = C
must be positive.
In this form, m = -A/B and the intercepts are (0,C/B) and (C/A, 0).
Every line has an equation that can be written in the standard form
Ax + By = C
must be positive.
In this form, m = -A/B and the intercepts are (0,C/B) and (C/A, 0).
Standard Form of the Equation of the a Line

Equations of Lines
• Point-slope form: y – y1 = m(x – x1)
• Slope-intercept form: y = m x + b
• Horizontal line: y = b
• Vertical line: x = a
• General form: Ax + By + C = 0
• Standard form: Ax + By = C
• Point-slope form: y – y1 = m(x – x1)
• Slope-intercept form: y = m x + b
• Horizontal line: y = b
• Vertical line: x = a
• General form: Ax + By + C = 0
• Standard form: Ax + By = C

Example: Finding the Slope and the y-Intercept
Find the slope and the y-intercept of the line whose equation is 2x – 3y + 6 =
0.
Solution The equation is given in general form, Ax + By + C = 0. One
method is to rewrite it in the form y = mx + b. We need to solve for y.
This is the given equation.2x – 3y + 6 = 0
To isolate the y-term, add 3 y on both sides.2x + 6 = 3y
Reverse the two sides. (This step is optional.)3y = 2x + 6
The coefficient of x, 2/3, is the slope and the constant term, 2, is the y-
intercept.
Divide both sides by 3.
2
2
3
y x= +

Steps for Graphing y = mx + b
Graphing y = mx + b by Using the Slope and y-Intercept
• Plot the y-intercept on the y-axis. This is the point (0, b).
• Obtain a second point using the slope, m. Write m as a
fraction, and use rise over run starting at the y-intercept to
plot this point.
• Use a straightedge to draw a line through the two points.
Draw arrowheads at the ends of the line to show that the
line continues indefinitely in both directions.
Graphing y = mx + b by Using the Slope and y-Intercept
• Plot the y-intercept on the y-axis. This is the point (0, b).
• Obtain a second point using the slope, m. Write m as a
fraction, and use rise over run starting at the y-intercept to
plot this point.
• Use a straightedge to draw a line through the two points.
Draw arrowheads at the ends of the line to show that the
line continues indefinitely in both directions.

Example: Graphing by Using Slope and
y-Intercept
Graph the line whose equation is y = x + 2.
Solution The equation of the line is in the form y = mx + b.
We can find the slope, m, by identifying the coefficient of x.
We can find the y-intercept, b, by identifying the constant term.
2
3
y = x + 22
3
The slope is
2/3.
The y-intercept
is 2.
moremore

Example: Graphing by Using Slope and y-Intercept
Graph the line whose equation is y = x + 2.2
3
We plot the second point on the line by starting at
(0, 2), the first point.
Then move 2 units up (the rise) and 3 units to the right
(the run). This gives us a second point at (3, 4).
Solution
We need two points in order to graph the line. We can use the y-intercept, 2,
to obtain the first point (0, 2). Plot this point on the y-axis.
2 Rise
.
3 Run
m = =
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5

Sample Problems
Give the slope and y-intercept of the given line then
graph.
3 2
2
- 6
5
y x
y x
= +
= +

Example: Finding the slope and the x-&y-intercepts.
Find the slope and the intercepts of the line whose equation
is 2x – 3y = -6.
Solution When an equation is given in standard form, Ax + By = C, the
slope can be determine by using the coefficients A and B, so that m = -A/B.
2x – 3y = -6 For the given equation, A = 2 and B = -3. So m = 2/3.
To find the intercepts, recall that the x-intercept has the form (x,0) and the
y-intercept has the form (0,y).
2x – 3(0) = -6 Let y = 0 and solve for x.
2x = -6
x = -3 So the x-intercept is (-3,0).
2(0) – 3y = -6 Likewise, let x = 0 and solve for y.
-3y = -6
y = 2 So the y-intercept is (0,2).

Problems
For the given equations,
1. Rewrite the equation in slope-intercept form and in
standard form.
2. Graph the lines using both methods – using slope and y-
intercept and using the x- & y-intercepts.
• 4x + y – 6 = 0
• 4x + 6y + 12 = 0
• 6x – 5y – 20 = 0
• 4y + 28 = 0
Exercises page 138, numbers 1-60.

Section 1.2 (cont’d)
Review
• Defintion of a slope :
• 6 Forms for the Equation of a Line
– Point-slope form: y – y1 = m(x – x1)
– Slope-intercept form: y = m x + b
– Horizontal line: y = b
– Vertical line: x = a
– General form: Ax + By + C = 0
– Standard form: Ax + By = C
• Graphing Techniques
– Using slope and y-intercept
– Using x- & y-intercepts
m=
y2 −y1
x2 −x1

Slope and Parallel Lines
• If two non-vertical lines are parallel, then they have the same slope.
• If two distinct non-vertical lines have the same slope, then they are
parallel.
• Two distinct vertical lines, both with undefined slopes, are parallel.
• If two non-vertical lines are parallel, then they have the same slope.
• If two distinct non-vertical lines have the same slope, then they are
parallel.
• Two distinct vertical lines, both with undefined slopes, are parallel.

Example: Writing Equations of a Line Parallel
to a Given Line
Write an equation of the line passing through (-3, 2) and parallel to the
line whose equation is y = 2x + 1. Express the equation in point-slope
form and y-intercept form.
Solution We are looking for the equation of the line shown on the left
on the graph. Notice that the line passes through the point (-3, 2). Using
the point-slope form of the line’s equation, we have x1 = -3 and y1 = 2.
y = 2x + 1
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
(-3, 2)
Rise = 2
Run = 1
y – y1 = m(x – x1)
y1 = 2 x1 = -3
moremore

Since parallel lines have the same slope and the slope of the given line is
2, m = 2 for the new equation. So we know that m = 2 and the point (-3,
2) lies on the line that will be parallel. Plug all that into the point-slope
equation for a line to give us the line parallel we are looking for.
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
(-3, 2)
Rise = 2
Run = 1
y = 2x + 1
moremore
Example continued:
y – y1 = m(x – x1)
y1 = 2 m = 2 x1 = -3

Example continued:
Solution The point-slope form of the line’s equation is
y – 2 = 2[x – (-3)]
y – 2 = 2(x + 3)
Solving for y, we obtain the slope-intercept form of the equation.
y – 2 = 2x + 6 Apply the distributive property.
y = 2x + 8 Add 2 to both sides. This is the slope-intercept
form of the equation.

Slope and Perpendicular Lines
• If two non-vertical lines are perpendicular, then the product of their
slopes is –1.
• If the product of the slopes of two lines is –1, then the lines are
perpendicular.
• A horizontal line having zero slope is perpendicular to a vertical line
having undefined slope.
• If two non-vertical lines are perpendicular, then the product of their
slopes is –1.
• If the product of the slopes of two lines is –1, then the lines are
perpendicular.
• A horizontal line having zero slope is perpendicular to a vertical line
having undefined slope.
Two lines that intersect at a right angle (90°) are
said to be perpendicular. There is a relationship
between the slopes of perpendicular lines.
90°

Example: Finding the Slope of a Line
Perpendicular to a Given Line
Find the slope of any line that is perpendicular to the line whose equation
is x + 4y – 8 = 0.
Solution We begin by writing the equation of the given line in slope-
intercept form. Solve for y.
x + 4y – 8 = 0 This is the given equation.
4y = -x + 8
To isolate the y-term, subtract x and add 8 on
both sides.
Slope is –1/4.
y = -1/4x + 2 Divide both sides by 4.
The given line has slope –1/4. Any line perpendicular to this line has a slope
that is the negative reciprocal, 4.

Write the equation of the line perpendicular to x + 4y – 8 = 0 that
passes thru the point (2,8) in standard form.
Solution: The given line has slope –1/4. Any line perpendicular to
this line has a slope that is the negative reciprocal, 4.
So now we need know the perpendicular slope and are given a
point (2,8). Plug this into the point-slope form and rearrange
into the standard form.
y – y1 = m(x – x1)
y1 = 8 m = 4 x1 = 2
y – 8 = 4[x – (2)]
Example: Writing the Equation of a Line
Perpendicular to a Given Line
y - 8 = 4x - 8
-4x + y = 0
4x – y = 0 Standard form

Problems
1. Find the slope of the line that is
a) parallel
b) perpendicular to the given lines.
• y = 3x
• 8x + y = 11
• 3x – 4y + 7 = 0
• y = 9
2. Write the equation for each line in slope-intercept form.
• Passes thru (-2,-7) and parallel to y = -5x+4
• Passes thru (-4, 2) and perpendicular to
y = x/3 + 7
Exercises pg 138, numbers 61-68

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