Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
121
6.36 Express the linear equation 6y = 5x + 2(x + 10) – 16 in standard form.
Expand and simplify the right side of the equation.
Subtract 7x from both sides of the equation and multiply all the terms by –1 so
that the equation has form Ax + By = C, where A > 0.
Creating Linear Equations
Practice all the skills from this chapter
Note:Problems 6.37–6.39 refer to line j, which passes through point (3,–1) and is parallel to
line p, which has equation .
6.37 Apply the point-slope formula to determine the equation of line j.
Line p is in slope-intercept form, so the x-coefficient is the slope of line p.
The slopes of parallel lines are equal, so the slope of line j must be as well.
Substitute and the coordinates of the point through which j passes
(x
1
= 3 and y
1
= –1) into the point-slope formula.
You could
subtract 6y
and 4 from both
sides of the equation
instead to get –4 =
7x – 6y. Then you’d
swap the sides of the
equation to get
7x – 6y = –4.
This property of
parallel lines was
rst introduced
back in Problem
5.33.
Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
122
Note: Problems 6.37–6.39 refer to line j, which passes through point (3,–1) and is parallel to
line p, which has equation .
6.38 Express line j in slope-intercept form.
Expand the right side of the point-slope equation generated in Problem 6.37
and solve for y.
Note: Problems 6.37–6.39 refer to line j, which passes through point (3,–1) and is parallel to
line p, which has equation .
6.39 Express line j in standard form.
Multiply each term of the equation in slope-intercept form (from Problem 6.38)
by 3 to eliminate the fraction.
Subtract 5x from both sides of the equation and multiply both sides by –1 to
rewrite the equation in form Ax + By = C, such that A > 0.
Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
123
Note: Problems 6.40–6.42 refer to line k, which passes through point (–8,7) and is
perpendicular to the line l, which has equation 2x – 3y = 12.
6.40 Calculate the slope of line k.
Calculate the slope of line l by applying the slope shortcut formula from
Problem 6.30.
Line k is perpendicular to line l, so the slope of k is , the opposite reciprocal
of line l’s slope.
Note: Problems 6.40–6.42 refer to line k, which passes through point (–8,7) and is
perpendicular to the line l, which has equation 2x – 3y = 12.
6.41 Use the slope-intercept formula to create the equation of line k.
Line k passes through point (–8,7) and, according to Problem 6.40, has slope
. Substitute , x = –8, and y = 7 into the slope-intercept formula in
order to calculate b.
Apply the slope-intercept formula to generate the equation of line k.
Note: Problems 6.40–6.42 refer to line k, which passes through point (–8,7) and is
perpendicular to the line l, which has equation 2x – 3y = 12.
6.42 Express line k in standard form.
According to Problem 6.41, the equation of line k in slope-intercept form is
. Multiply all the terms by the least common denominator (2) and
add the x-term to both sides of the equation. The resulting equation has form
Ax + By = C such that A > 0.
The slopes
of perpendic-
ular lines are
opposites—if one
is positive then the
other is negative—
and reciprocals of
each other. If you
feel like you heard
that somewhere
before, you might
remember it
from Problem
5.35.
Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
124
6.43 Write the equation of the line graphed in Figure 6-6 in slope-intercept form.
Figure 6-6: You can determine the equation of this line using either point-slope or slope-
intercept form, but write your final answer in slope-intercept form.
The line in Figure 6-6 passes through points (–5,–1) and (4,3). Use the slope
formula to calculate the slope of the line.
Substitute the slope and the x- and y-values from one of the points through
which the line passes into the point-slope formula.
Expand the right side of the equation and solve for y to express the equation in
slope-intercept form.
Subtract
1 from both
sides of the
equation to isolate y
in this step. Instead of
“–1,” you can write
.
It’s the same value
(
), but you
need a denominator
in common with
so that you can add
those numbers
together.
Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
125
6.44 Write the equation of the line with x-intercept and y-intercept in
standard form.
Calculate the slope of the line that passes through points and .
The slope of the line is and the y-intercept is stated: . Substitute
these values into the slope-intercept formula to create the equation of the line.
Multiply each of the terms by the least common denominator (14) to eliminate
the fractions in the equation.
Subtract 10x from both sides of the equation and multiply each term by –1 so
that the equation has form Ax + By = C and A > 0.
To get rid of this
complex fraction,
multiply the numerator
and denominator by
2, the least common
denominator.
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