Chapter Five — Graphing Linear Equations in Two Variables
The Humongous Book of Algebra Problems
93
Note: Problems 5.24–5.25 refer to the equation 5x = 3y + 16.
5.25 Graph the equation using the intercepts calculated in Problem 5.24.
Calculate the mixed number equivalents of the intercepts identified in Problem
5.24 and , plot the points, and connect them to draw the
graph, which is illustrated in Figure 5-15.
Figure 5-15: The graph of 5x = 3y + 16 passes through points and .
Calculating Slope of a Line
Figure out how slanty a line is
5.26 The slope of a line is defined as . Explain how to calculate the slope
of a line that passes through points (x
1
,y
1
) and (x
2
,y
2
).
The slope of a line is the quotient of the vertical change of a line (the change
in the y direction) and the horizontal change of a line (the change in the x
direction). To calculate the slope of a line, identify two points on the line,
subtract their y-values, and divide by the difference of the corresponding x-
values. Here, a line passes through points (x
1
,y
1
) and (x
2
,y
2
), so the slope of the
line is .
The little
triangles are
the Greek letter
delta, which means
“change in” when it
comes to math. So the
slope is equal to the
change in y divided
by the change
in x.
The letter m
is usually used to
represent slope, even
though the word “slope”
doesn’t have an m in it.
Here’s another one that
boggles my mind: The
y-intercept of a line is
usually represented
by the variable b.
Go gure.
Chapter Five — Graphing Linear Equations in Two Variables
The Humongous Book of Algebra Problems
94
5.27 Describe the difference between a linear graph with a positive slope and a
graph with a negative slope.
As illustrated by Figure 5-16, lines with positive slopes rise from left to right.
As the x-values increase from negative to positive (as you travel right along the
x-axis), the y-values increase as well (the graph climbs vertically). On the other
hand, lines with negative slopes decline from left to right in the coordinate
plane.
Figure 5-16: Line k has a positive slope, so it increases from left to right in the
coordinate plane. Conversely, line l has a negative slope and decreases.
5.28 Calculate the slope of the line that passes through points (1,4) and (6,2).
To apply the formula from Problem 5.26 , set (1,4) = (x
1
,y
1
) and
(6,2) = (x
2
,y
2
). Substitute the values into the slope formula to determine the
slope of the line.
5.29 Calculate the slope of the line that passes through points (–5,9) and (1,–9).
Substitute x
1
= –5, y
1
= 9, x
2
= 1, and y
2
= –9 into the slope formula.
In other words,
x
1
= 1, y
1
= 4,
x
2
= 6, and y
2
= 2. If the
subscripts match, make
sure the x- and y-values
come from the same
coordinate pair.
Chapter Five — Graphing Linear Equations in Two Variables
The Humongous Book of Algebra Problems
95
5.30 Calculate the slope of the line that passes through points and
.
Substituting these values into the slope formula produces a complex fraction.
To simplify the numerator and denominator of the slope, ensure that the
fractions you combine have common denominators.
Once the numerator and denominator are rational numbers, rewrite the
fraction as a quotient and simplify.
5.31 Calculate the slope of the horizontal line y = 2.
To calculate slope using the formula m = , you need two points, (x
1
,y
1
)
and (x
2
,y
2
), on the line. Every point on the line y = 2 has a y-value of 2; no
matter what real number is used for the x-value, the point (x,2) belongs to the
horizontal line. For instance, set x
= 0 and x = 5 to get points (0,2) and (5,2) on
the graph of y = 2. Apply the slope formula.
The slope of the line y = 2 is 0.
For more
information
about simplifying
complex fractions,
see Problems 2.41–
2.43.
A fraction
over a fraction
is just a division
problem, and dividing
is the same thing
as multiplying by a
reciprocal. That’s how
you change
into
.
Every horizontal line has slope 0. All the points on
the line have the same y-value, and when you subtract
equal y-values in the numerator of the slope formula, you get
0. Zero divided by any real number except zero will equal 0.
Chapter Five — Graphing Linear Equations in Two Variables
The Humongous Book of Algebra Problems
96
5.32 Calculate the slope of the vertical line x = c. Assume that c is a real number.
All coordinate pairs on the line x = c have an x-value of c. Any value of y can be
used to complete the coordinate pairs. If a and b are distinct real numbers, then
(c,a) and (c,b) belong to the graph of x = c. Apply the slope formula to calculate
the slope.
Division by zero produces an undefined result. The slope of line x = c, like the
slope of any vertical line, is undefined. It is equally correct to say that the line
has “no slope.” However, it is incorrect to state that x = c has zero slope, because
the number zero is defined. Horizontal lines have zero slope, and vertical lines
have no slope (or an undefined slope).
5.33 If line k in the coordinate plane has slope and line l is parallel to line k,
what is the slope of l?
If two lines are parallel, the slopes of those lines are equal. Therefore, line l also
has slope .
5.34 Assume s and t are parallel lines. If line s passes through points (0,–2) and
(–5,12) and line t has x-intercept 3, what is the y-intercept of line t?
If s and t are parallel lines, their slopes are equal. Use the given points to
calculate m
s
, the slope of line s.
The slope of line s (and, therefore, the slope of line t) is . If line t has
x-intercept 3, then it passes through the point (3,0). Let (0,y) be the y-intercept
of line t. In this instance, the slope is already known . Apply the
slope formula again, substituting x
1
= 3, y
1
= 0, x
2
= 0, and y
2
= y. Solve the
resulting proportion for y.
If a and
b are equal, you
get 0 divided by 0,
which is a topic for
another day. For now,
let’s just say a can’t
equal b.
If you
don’t like
using the
abstract y-
values a and b,
you don’t have to.
You can pick any
real numbers, say for
example y = 1 and
y = 4, instead. The
slope of the line
through (c,1) and
(c,4) is also
undened.
Remember,
the x-value of
a y-intercept is 0,
just like the y-value
of an x-intercept
is 0.
A proportion
is an equation
with one fraction on
each side. You usually
use cross multiplication
to solve proportions,
and that process is
explained in Problems
21.1–21.8.
You should multiply
both sides of this equation
by –1 to cancel out the
negative signs; that’s why
they’re gone in the next
step.
Chapter Five — Graphing Linear Equations in Two Variables
The Humongous Book of Algebra Problems
97
5.35 If line k has slope and line l is perpendicular to line k, what is the slope of l?
The slopes of perpendicular lines are opposite reciprocals. In other words, the
slopes are reciprocals and have opposite signs. If the slope of line k is , then
the slope of line l must be .
5.36 Assume s and t are perpendicular lines. Line s passes through point (–6,–2)
and line t passes through point (x,8); the lines intersect at (–5,1). Find x.
Lines s and t intersect at (–5,1), so both lines pass through that point.
Therefore, line s contains the points (–6,–2) and (–5,1). Calculate m
s
, the slope
of line s.
Because s and t are perpendicular lines, their slopes are opposite reciprocals,
so if m
s
= , then m
t
= . Recall that line t passes through points (–5,1) and
(x,8). Apply the slope formula.
Cross multiply and solve the proportion for x.
5.37 Calculate the x- and y-intercepts of the linear equation Ax + By = C, and use
them to generate a formula for the slope of a line written in that form. Assume
that A, B, and C are real numbers and .
Calculate the x-intercept by substituting y = 0 into the equation and calculate
the y-intercept by substituting x = 0 into the equation.
- -
If A, B, and C
aren’t fractions
and A is positive,
then a line in the
form Ax + By = C is
in “standard form.”
More on this in
Problems 6.29–
6.36.
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