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The Humongous Book of Algebra Problems by W. Michael Kelley

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Chapter Five — Graphing Linear Equations in Two Variables
The Humongous Book of Algebra Problems
93
Note: Problems 5.245.25 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
doesnt have an m in it.
Heres 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.

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