The Humongous Book of Algebra Problems

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.