2
3
x
+ 3
y
= 1
222 Chapter 8: Systems of Equations
32. Let w,x,y, and zrepresent the ages of Tammy, Carmen,
Dennis, and Mark respectively. Solve:
Chapter 8 Vocabulary Reinforcement
Chapter 8 Concept Reinforcement
2. True; see page 565 in the text.
Chapter 8 Study Guide
x+3y=1 x+y=3
4+3(−1)?1 4+(−1)?3
2. 2x+y=2,(1)
3x+2y= 5 (2)
Substitute −1 for xin Equation (3) and find y.
y=−2(−1)+2=2+2=4
3. 2x+3y=5,(1)
Substitute −2 for xin one of the original equations and
solve for y.
4. Familiarize. We let xand yrepresent the amounts of
4
5
x
–
y
= –3
—2
Chapter 8 Summary and Review: Review Exercises 223
Translate. We organize the information in a table.
Time 1 yr 1 yr
Interest 0.06x0.05y$1237
The “Principal” row of the table gives us one equation:
After clearing decinmals, we have the following system of
equations.
Substitute 8700 for xin Equation (1) and solve for y.
8700 + y=23,000
5. x−y+z=9,(1)
Now solve the system of Equations (4) and (5).
2·3+y+2·1 = 3 Substituting 3 for x
Chapter 8 Review Exercises
The solution (point of intersection) seems to be the point
(−2,1).
The solution is (−2,1).
1
2
3. Graph both lines on the same set of axes.
independent.
4. 2x−3y=5,(1)
5. y=x+2,(1)
y−x= 8 (2)
x+2−x=8
6. 7x−4y=6,(1)
y−3x=−2 (2)
Next substitute 2
5for xin Equation (3) and find y.
7. x+3y=−3,(1)
8. 3x−5y=−4,(1)
5x−3y= 4 (2)
6−5y=−4
9. 1
3x+2
9y=1,
by 2 on both sides of the second equation to clear the
fractions.
10. 1.5x−3=−2y,
3x+4y=6
11. Familiarize. Let s= the speed of the plane, in mph, in
still air and d= the distance traveled each way, in miles.
d=(s+ 30)3,(1)
d=(s−30)4.5 (2)
Solve. We substitute (s+ 30)3 for din Equation (2).
12. Familiarize. Let x= the number of less expensive
brushes sold and y= the number of more expensive
brushes sold.
We have a system of equations:
x+y=45,
y=13
13. Familiarize. Let xand yrepresent the number of liters
of Orange Thirst and Quencho that should be used, re-
x+y=10
The last row yields a second equation.
226 Chapter 8: Systems of Equations
14. Familiarize. We first make a drawing.
train d44 t+1 →d=
44(t+1)
Fast
a system of equations:
d= 44(t+1),
d=52t
Solve. We solve the system of equations.
15. x+2y+z=10,(1)
2x−y+z=8,(2)
z=−8
5x+3(−8) = 26 Substituting −8 for zin (4)
16. 3x+2y+z=1,(1)
We start by eliminating xfrom two different pairs of equa-
tions.
3x+2y+z= 1 (1)
−3x+9y+6z= 18 Multiplying (3) by 3
−x+3·3+2(−2) = 6 Substituting 3 for yand
−2 for zin (3)
Chapter 8 Summary and Review: Review Exercises 227
17. 2x−5y−2z=−4,(1)
−2x+3y+2z= 4 (3)
−2y=0
y=0
Substitute 0 for yin Equation (4) and solve for z.
18. x+y+2z=1,(1)
x−y+z=1,(2)
−x=−2
x=2
2+y+2
−2
19. Familiarize. Let a,b, and crepresent the measures of
angles A, B, and C, respectively. Recall that the sum of
the measures of the angles of a triangle is 180◦.
Translate.
We have a system of equations.
a+b+c=180,
20. Familiarize. Let x,y, and zrepresent the prices of 1 bag
of caramel nut crunch popcorn, 1 bag of plain popcorn,
x
1—1—33254—2—4—5
—1
—2
1
2
3
(0, 2)
(1, 3)
228 Chapter 8: Systems of Equations
We have a system of equations.
x+y+z=49,
21. x−y=−9,(1)
y−2x= 9 (2)
Substitute 0 for xin Equation (2) and solve for y.
y−2·0=9
22. Familiarize. Let xand yrepresent the numbers.
Translate.
The sum of the numbers
is −2.
x+y=−2,(1)
2x+y= 4 (2)
Solve. We multiply both sides of Equation (1) by −1 and
then add.
23. Familiarize. Let t= the time the cars travel, and let d=
Distance Rate Time
275 = 110t
2.5=t
Check. In 2.5 hr, the first car travels 50(2.5), or 125 mi,
24. We graph the equations and find the points of intersection.
Chapter 8 Discussion and Writing Exercises
1. Answers may vary. One day a florist sold a total of 23
hanging baskets and flats of petunias. Hanging baskets
tion (2) is 10x+3y+0.5z= 100. There is no other in-
formation that can be translated to an equation. Clearing
y
—5
x
1—1—33254—2—4—5
—1
—2
—3
—4
1
2
3
—5
6
x
– 2
y
= –12
Chapter 8 Test 229
Since we have only two equations, it is not possible to
Chapter 8 Test
1. Graph both lines on the same set of axes.
The solution (point of intersection) seems to be the point
2. Graph both lines on the same set of axes.
4. 4x+3y=−1,(1)
y=2x−7 (2)
230 Chapter 8: Systems of Equations
5. x=3y+2,(1)
2x−6y= 4 (2)
Substitute 3y+ 2 for xin the second equation and solve
The system of equations has infinitely many solutions.
6. x+2y=6,(1)
2x+3y= 7 (2)
y=5
7. t=2−r, (1)
3r−2t= 36 (2)
Now substitute 8 for rin Equation (1) to find t.
8. 2x+5y=3,(1)
−2x+3y= 5 (2)
The ordered pair (−1,1) checks in both equations. It is
the solution.
9. x+y=−2,(1)
x=−3
2
Now substitute −3
10. 2
151
3x−2
5y=15·2→5x−6y= 30 (4)
11. 0.3a−0.4b=11,
3a−4b=110,(1)
7a+12b=−170 (2)
Chapter 8 Test 231
b=−20
12. Familiarize. Let l= the length and w= the width, in
feet.
13. Familiarize. Let d= the distance traveled and r= the
speed of the plane in still air, in km/h. Then with a 20-
km/h tailwind the plane’s speed is r+ 20. The speed of
14. Familiarize. Let b= the number of buckets of wings and
Buckets Dinners Total
Number
sold b d 28
sold.
15. Familiarize. Let x= the number of liters of 20% solution
and y= the number of liters of 45% solution to be used.
232 Chapter 8: Systems of Equations
Solve. We use the elimination method.
−20x−20y=−400 Multiplying (1) by −20
20x+45y= 600
16. 6x+2y−4z=15,(1)
18x+6y−12z= 45 Multiplying (1) by 3
4x−6y+3z= 8 (3)
22x−9z= 53 (5)
Now solve the system of Equations (4) and (5).
9x−6z= 24 (4)
17. Familiarize. Let x,y, and zrepresent the number of
hours worked by the electrician, the carpenter, and the
Translate.
Total time worked
is 21.5 hr.
21x+19.50y+24z= 469.50,
z=2+y
18. Familiarize. Let x,y, and zrepresent the amounts in-
vested at 2%, 3%, and 5%, respectively. The amounts
earned by the funds are 0.02x,0.03y, and 0.05z.
Translate.
Amount invested
was $30,000.
↓↓↓
2%. The answer checks.
State. $10,000 was invested at 5%.
19. Substituting −1 for xand 3 for f(x), we have
7= m
Cumulative Review Chapters 1 – 8
1. (3x4−2y5)(3x4+2y5)=(3x4)2−(2y5)2=9x8−4y10
4. x
2x−1−3x+2
1−2x=x
2x−1−3x+2
1−2x·−1
−1
6. 2x+2
3x−9·x2−8x+15
x2−1=(2x+ 2)(x2−8x+15)
(3x−9)(x2−1)
7. 2x2−2
2x2+7x+3÷4x−4
2x2−5x−3
8. 3x2+4x+9
13
The answer is 3x2+4x+9+ 13
x−2.
9. 3−12x8= 3(1 −4x8)
=4x2(x+1)−(x+1)
=(x+ 1)(4x2−1)
16. y=mx +b
2=−3·5+bSubstituting
The slope of the given line is 1
3(x+6)
18. First we write each equation in slope-intercept form.
x−2y=4 4x+2y=1
lines are perpendicular.
19. x2=−17x
20. 1
4x+2
3x=2
3−3
4x, LCM is 12
2
x=2
21. 1
x+2
3=1
4,LCM=12x
22. x2−30 = x
−6x−20 ≥7
−6x≥27
24. x
x−1−x
x+1 =1
2x−2
2(x−1)(x+1)·x
x+1 =x+1
2x(x+1)−2x(x−1)=x+1
25. 4A=pr +pq
r+q=p
26. 3x+4y=4,(1)
x
3y 9
y
Cumulative Review Chapters 1 – 8 235
27. 3x+y=2,(1)
9x= 9 Adding
28. 4x+3y=5,(1)
3x+2y= 3 (2)
29. x−y+z=1,(1)
2x+y+z=3,(2)
3x+2z= 4 (4)
3x+2z= 4 (4)
4x−2z= 10 Multiplying (5) by 2
y=0
30. 3y=9
32. 3x−1=y
Find some ordered pairs that are solutions, plot these
236 Chapter 8: Systems of Equations
33. 3x+5y=15
The y-intercept is (0,3).
34. a) The domain is the set of all x-values in the graph,
{−5,−3,−1,1,3}.
35. f(x)= 7
2x−1
36. g(x)=1−2x2
37. Familiarize. Let x= the number of liters of 15% solution
15%
25%
of alcohol 15% 25% 18%
−15x−15y=−450
15x+25y= 540
Check. The mixture contains 21 L + 9 L, or 30 L. The
amount of alcohol in the mixture is 0.15(21) + 0.25(9), or
3.15L+2.25, or 5.4 L. The answer checks.
Electric bill
was one-fourth of the rent.
Cumulative Review Chapters 1 – 8 237
39. Familiarize. Let d= the number of defective resistors
you would expect to find in a sample of 250.
150 =d
250
250 ·12
40. Familiarize. Let w= the width of the rectangle, in me-
(w+ 15)(w−12)=0
w+15= 0 or w −12=0
w=−15 or w =12
41. Familiarize and Translate. We have inverse variation
between Aand d, so an equation A=k/d,k>0, applies.
A=400
1000
A=0.4
42. Familiarize. Let x= the measure of the first angle. Then
2x= the measure of the second angle and x+2x−48, or
6x−48=180
43. We find a function that relates the amount spent on radio
advertising to weekly sales increase. We have the data
points (1000,101,000) and (1250,126,000).
Then we have the function
S(a) = 100a+ 1000
44. We have the points (5,−3) and (−4,2).
m=−3−2
5−(−4) =−5
9=−5
9