Chapter 5
34. ∣ 3x – 45 ∣ = ∣ 12x ∣
3x – 45 = 12x
–3x
or
–3x
Check ∣ 2x + 1 ∣ = ∣ 3x – 11 ∣
?
∣ 2( 12 ) + 1 ∣ = ∣ 3( 12 ) – 11 ∣
?
∣ 24 + 1 ∣ = ∣ 36 – 11 ∣
?
∣ 25 ∣ = ∣ 25 ∣
3x – 45 = –12x
–3x
–3x
–45 = 9x
–45 = –15x
–45 9x
—=—
9
9
–5 = x
–45 –15x
—=—
–15
–15
3=x
Check ∣ 3x – 45 ∣ = ∣ 12x ∣
?
∣ 3( 3 ) – 45 ∣ = ∣ 12( 3 ) ∣
?
∣ –15 – 45 ∣ = ∣ –60 ∣
?
∣ –60 ∣ = 60
?
∣ 9 – 45 ∣ = ∣ 36 ∣
?
∣ –36 ∣ = 36
36 = 36 ✓
The solutions are x = –5 and x = 3.
35. ∣ x – 7 ∣ = ∣ 2x – 8 ∣
x – 7 = 2x – 8
–x
x – 7 = −( 2x – 8 )
or
–x
x – 7 = –2x + 8
–7 = x – 8
+8
+2x
+8
+2x
3x – 7 = 8
1=x
+7
+7
3x = 15
3x 15
3
3
x=5
—=—
Check ∣ x – 7 ∣ = ∣ 2x – 8 ∣
?
∣ 1 – 7 ∣ = ∣ 2( 1 ) – 8 ∣
?
∣ –6 ∣ = ∣ 2 – 8 ∣
?
6 = ∣ –6 ∣
∣ x – 7 ∣ = ∣ 2x – 8 ∣
?
∣ 5 – 7 ∣ = ∣ 2( 5 ) – 8 ∣
?
∣ –2 ∣ = ∣ 10 – 8 ∣
2 = ∣2∣
6=6✓
2=2✓
The solutions are x = 1 and x = 5.
36. ∣ 2x + 1 ∣ = ∣ 3x –11 ∣
2x + 1 = 3x – 11
–2x
2x + 1 = – ( 3x – 11 )
or
–2x
1 = x – 11
+11
2x + 1 = –3x + 11
+3x
+11
12 = x
+3x
5x + 1 = 11
–1
–1
5x = 10
5x
5
10
5
—=—
x=2
266
Algebra 1
Worked-Out Solutions
?
∣ 4 + 1 ∣ = ∣ 6 – 11 ∣
?
∣ 5 ∣ = ∣ –5 ∣
5=5✓
The solutions are x = 12 and x = 2.
∣ 3( –5 ) – 45 ∣ = ∣ 12( –5 ) ∣
60 = 60 ✓
?
∣ 2( 2 ) + 1 ∣ = ∣ 3( 2 ) – 11 ∣
25 = 25 ✓
∣ 3x – 45 ∣ = ∣ 12x ∣
?
∣ 2x + 1 ∣ = ∣ 3x – 11 ∣
5.1–5.4 What Did You Learn? (p. 259)
1. You know the total number of songs played, the relationship
between the number of pop songs played and the number
of rock songs played, and the relationship between the
number of hip-hop songs played and the number of rock
songs played. The solution can be found by writing a system
of three linear equations in three variables that represents
the problem. Then, substitute an expression for x and an
expression for z into the first equation that contains all three
variables. Solve this equation for y. Substitute the value of
y into each of the other two equations and solve for x and z,
respectively.
2. Sample answer: An Internet site offers commercial-free
viewing of individual episodes of a TV show for one price
or access to an entire season of a TV show for another price.
If you knew how many individual shows and seasons you
purchased in Month 1 and Month 2 and the total charge for
each of those months, you could write a system of linear
equations, similar to the one in Exercise 22, that could be
solved to find the cost of viewing one episode and the cost
for access to an entire season.
3. Sample answer: What are the slope and y-intercept of the
line that describes the first receipt? the second receipt? How
are these two equations related? What does that tell you
about the system of linear equations?
5.1–5.4 Quiz (p. 260)
1. The lines appear to intersect at (3, 1).
1
Check y = −—3 x + 2
? 1
1 = −—3 (3) + 2
?
1 = −1 + 2
y=x−2
?
1=3−2
1=1✓
1=1✓
The solution is (3, 1).
2. The lines appear to intersect at (−2, −2).
Check
y = —12 x − 1
y = 4x + 6
? 1
−2 = —2 (−2) − 1
?
−2 = −1 − 1
?
−2 = 4(−2) + 6
?
−2 = −8 + 6
−2 = −2 ✓
−2 = −2 ✓
The solution is (−2, −2).
Copyright © Big Ideas Learning, LLC
All rights reserved.
Chapter 5
3. The lines appear to intersect at (0, 1).
Check y = 1
1=1✓
x + 4y = 10
6. Step 1
y = 2x + 1
?
1 = 2(0) + 1
?
1=0+1
x + 4y − 4y= 10 − 4y
x = 10 − 4y
3x − 5y = 13
Step 3 x + 4y = 10
3(10 − 4y) − 5y = 13
x + 4(1) = 10
3(10) − 3(4y) − 5y = 13
x + 4 = 10
Step 2
1=1✓
The solution is (0, 1).
30 − 12y − 5y = 13
4. Substitute x − 4 for y in Equation 2 and solve for x.
−2x + y = 18
30 − 17y = 13
Step 3 y = x − 4
−2x + (x − 4) = 18
−30
y = −22 − 4
−2x + x − 4 = 18
−17y −17
−17
−17
y=1
+4
Check
y=x−4
?
−26 = −22 − 4
−2x + y = 18
?
−2(−22) + (−26) = 18
?
44 − 26 = 18
−26 = −26 ✓
13 = 13 ✓
7. Step 2
Step 3
y − x + x = −5 + x
y=x−5
2y + x = −4
Step 3 y − x = −5
2(x − 5) + x = −4
y − 2 = −5
+2
2x − 10 + x = −4
+10
3x = 6
3x 6
3
3
x=2
2y + x = −4
?
2(−3) + 2 = −4
?
−6 + 2 = −4
x+y=4
Step 4
−2x = −4
−2x −4
—=—
−2
−2
x=2
x+y=4
2+y=4
−2
−2
y=2
Check x + y = 4
−3x − y = −8
?
2+2=4
?
−3(2) − 2 = −8
4=4✓
—=—
Check
+2
y = −3
3x − 10 = −4
+10
10 = 10 ✓
−2x + 0 = −4
y − x = −5
2(x) − 2(5) + x = −4
x + 4y = 10
?
6 + 4(1) = 10
?
6 + 4 = 10
−3x − y = −8
The solution is (−22, −26).
Step 2
3x − 5y = 13
?
3(6) − 5(1) = 13
?
18−5 = 13
The solution is (6, 1).
18 = 18 ✓
5. Step 1
x=6
−30
—=—
−x = 22
22
−x
—=—
−1 −1
x = −22
Check
−4
−17y = −17
y = −26
−x − 4 = 18
+4
−4
?
−6 − 2 = −8
−8 = −8 ✓
y − x = −5
?
−3 −2 = −5
The solution is (2, 2).
−5 = −5 ✓
−4 = −4 ✓
The solution is (2, −3).
Copyright © Big Ideas Learning, LLC
All rights reserved.
Algebra 1
Worked-Out Solutions
267
Chapter 5
8. Step 1
11. Solve by elimination.
Step 2
x + 3y = 1
−2x − 6y = −2
Multiply by −2.
5x + 6y = 14
Step 1 6x + 2y = 16
5x + 6y = 14
2x − y = 2
Step 2 6x + 2y = 16
4x − 2y = 4
Multiply by 2.
3x + 0 = 12
Step 4 x + 3y = 1
Step 3
4 + 3y = 1
−4
−4
3x = 12
3x 12
—=—
3
3
x=4
3y = −3
3y −3
—=—
3
3
y = −1
Check
10x 20
10
10
x=2
—=—
12 + 2y = 16
−12
2y = 4
x + 3y = 1
?
4 + 3(−1) = 1
?
4−3=1
5x + 6y = 14
?
5(4) + 6(−1) = 14
?
20 − 6 = 14
14 = 14 ✓
2y 4
—=—
2
2
y=2
The solution is (2,2).
12. Solve by elimination.
Step 1
Step 2
3x − 3y = −2
The solution is (4, −1).
Multiply by 2.
−6x + 6y = 4
9. Step 1
6x − 6y = −4
−6x + 6y = 4
Step 2
2x − 3y = −5
Multiply by 2.
5x + 2y = 16
Multiply by 3.
0=0
4x − 6y = −10
15x + 6y = 48
19x + 0 = 38
5x + 2y = 16
Step 3
5(2) + 2y = 16
10 + 2y = 16
−10
19x = 38
19x 38
—=—
19
19
x=2
−10
The equation 0 = 0 is always true. So, the solutions are
all points on the line 3x − 3y = −2. The system of linear
equations has infinitely many solutions.
13. a. Words
14
+ 4
8
+
2y = 6
2y
2
6
2
y=3
2x − 3y = −5
?
2(2) − 3(3) = −5
?
4 − 9 = −5
6
⋅
Growing
time
(in years)
=
Height
(in inches)
⋅
Growing
time
(in years)
=
Height
(in inches)
Variables Let x be how long (in years) the trees are
growing, and let y be the height (in inches) of
the trees.
—=—
Check
10x = 20
Step 3
6(2) + 2y = 16
−12
1=1✓
Step 4
10x + 0 = 20
Step 4 6x + 2y = 16
5x + 2y = 16
?
5(2) + 2(3) = 16
?
10 + 6 = 16
−5 = −5 ✓
System 14 + 4x = y
8 + 6x = y
A system of linear equations that represents this situation
is y = 4x + 14 and y = 6x + 8.
16 = 16 ✓
The solution is (2, 3).
10. Solve by elimination.
Step 2
x−y=1
−(x − y = 6)
0 = −5
The equation 0 = −5 is never true. So, the system of linear
equations has no solution.
268
Algebra 1
Worked-Out Solutions
Copyright © Big Ideas Learning, LLC
All rights reserved.
Chapter 5
b.
Step 3 x + y = 3
Height (inches)
y
1+y=3
40
y = 4x + 14
−1
30
(3, 26)
20
10
0
−1
y=2
The solution is (1,2). So, you spend 1 hour driving at
55 miles per hour on highways, and you spend 2 hours
driving at 40 miles per hour on the rest of the roads.
y = 6x + 8
0
1
2
3
4
b. You drive 55x = 55(1) = 55 miles on highways and
x
40y = 40(2) = 80 miles on the rest of the roads.
Time (years)
Check
y = 4x + 14
?
26 = 4(3) + 14
?
26 = 12 + 14
y = 6x + 8
?
26 = 6(3) + 8
?
26 = 18 + 8
26 = 26 ✓
26 = 26 ✓
15. Words
7
The solution is (3, 26). So, in 3 years, both trees will be
26 inches tall.
14. a. Words
Time
(in hours)
on highway
+
Time
(in hours)
on other roads
=3
Number of
touchdowns
⋅
Number of
field goals
+
Number of
+ 3
touchdowns
⋅
=6
Number of
= 26
field goals
Variables Let x be the number of touchdowns the home
team scores, and let y be the number of field goals
the home team scores.
System x + y = 6
7x + 3y = 26
Solve by elimination.
55
⋅
Time
(in hours) + 40
on highway
⋅
Time
(in hours) = 135
on other roads
Variables Let x be how much time (in hours) you spend
driving at 55 miles per hour on highways, and
let y be how much time (in hours) you spend
driving at 40 miles per hour on the rest of
the roads.
System x + y = 3
Step 2
x+y=6
Multiply by −3.
7x + 3y = 26
−3x − 3y = −18
7x + 3y = 26
4x + 0 = 8
Step 4 x + y = 6
2+y=6
−2
−2
Step 3
4x = 8
4x 8
—=—
4
4
x=2
y=4
55x + 40y = 135
The solution is (2, 4). So, the home team scores
2 touchdowns and 4 field goals.
Solve by substitution.
Step 1
Step 1
x+y=3
x−x+y=3−x
y=3−x
Step 2
55x + 40y = 135
55x + 40(3−x) = 135
55x + 40(3) − 40(x) = 135
55x + 120 − 40x = 135
15x + 120 = 135
−120
−120
15x = 15
15x
15
15
15
—=—
x=1
Copyright © Big Ideas Learning, LLC
All rights reserved.
Algebra 1
Worked-Out Solutions
269