Chapter 8 Homework Adding Substitute For The Equation 2

x

y

1—1—3 3254—2—4—5

—1

x

y

1—1—33254—2—4—5

—1

(2, –1)

x

y

1—1—33254—2—4—5

—1

5

y

1—1—33254—2—4—5

—1

—2

2

3

4

5

f

(

x

) =

x

+ 1

g

(

x

) = –

x

2

(–3, –2)

b

—2

—4

—6

a

–

b

= 5

y

—2

—3

—4

y

5

Chapter 8

Systems of Equations

Exercise Set 8.1

RC2. True; see page 563 in the text.

2.

4.

6.

8.

10.

12.

14.

y

b

—2

—3

—4

a

+ 2

b

= –3

y

—2

—3

—4

5

x

–

y

= 0

206 Chapter 8: Systems of Equations

16.

18.

Consistent; independent

20.

22. Consistent; independent; the pair (1,1) is a solution of

both equations in system Cand is not a solution of any

y-intercept (0,−1) and a horizontal line. Both lines pass

26. Consistent; independent; the pair (−1,3) is a solution

28. 8y−3x= 2 Given line

y=mx +b

30. First we find the slope of the given line.

y=−3

4x+1

2

34.

Exercise Set 8.2

2. x=8−4y, (1)

−7y=−21

4. 9x−2y=3,(1)

9x−6x+12=3

6. m−2n=3,(1)

solve for n.

8. t=4−2s, (1)

4=6

10. 5x+6y=14,(1)

Substitute −1 for yin Equation (3).

12. 4p−2q=16,(1)

5p+7q= 1 (2)

19p−56=1

14. 3x+y=4,(1)

We have a true equation. Any value of xwill make this

16. 5x+3y=4,(1)

We solve the second equation for x.

x−4y= 3 (2)

208 Chapter 8: Systems of Equations

18. 4x+13y=5,(1)

The solution is (−2,1).

20. Let l= the length and w= the width. Solve:

22. Let xand yrepresent the angles. Solve:

x+y=90,

y=5x+6

and 21 first-class seats.

22 =x

32. 5x+2y=a,

2x−2y=2bMultiplying by 2

7x+0=a+2bAdding

34. Let b= the number of ounces of baking soda and v= the

number of ounces of vinegar to be used. Solve:

b=4v,

Exercise Set 8.3

RC2. If a system of equations has no solution, then it is

2. x+y=9,(1)

4. 2x−3y=18,(1)

Exercise Set 8.3 209

6. 2x+3y=−9,(1)

5x−6y=−9 (2)

4x+6y=−18 Multiplying (1) by 2

8. 2a+3b=11,(1)

−4a−6b=−22 Multiplying (1) by −2

Substitute 3 for bin (1).

10. 3x−2y=1,(1)

0 = 0 True for all (x, y)

There are infinitely many solutions.

12. 5x+4y=2,(1)

2x−8y= 4 (2)

The solution is 2

3,−1

3.

14. 5x+3y=25,(1)

y=5

16. 3x−5y=−2

18. 10x+y= 306,(1)

−99x=−2970

20. 2

3x+1

7y=−11,

1

7x−1

3y=−10

210 Chapter 8: Systems of Equations

22. 2

3x+3

5y=−17,

tion by 6 to clear fractions.

10x+9y=−255,(1)

24. 4

3x+3

2y=4,

5

20x−3y=−144 (2)

x=−6

26. 0.7x−0.3y=0.5,

−0.4x+0.7y=1.3

−4·2+7y=13

28. 1.3x−0.2y=12,(1)

0.4x+17y= 89 (2)

The solution is (10,5).

30. Let x= the larger number and y= the smaller number.

Solve:

32. Let x= the larger number and y= the smaller number.

Solve:

34. Let xand yrepresent the measures of the angles. Solve:

x+y=64,

2x+y=100

Exercise Set 8.4 211

38. Let x= the number of redbud trees chosen and y= the

number of dogwood trees chosen. Solve:

40. f(−1) = 3(−1)2−(−1)+1=3+1+1=5

46. m=10 −2

−2−(−10) =8

8=1

48. x+y

2−x−y

5=1,

6x+14y= 20 Multiplying (1) by 2

3−32

17+7y=10

17,38

17.

50. There are many correct answers. Find one by expressing

52. Substitute −4 for xand −3 for yin both equations and

−4b+3a= 7 (2)

−12a−9b=−78 Multiplying (1) by 3

Substitute 2 for bin (2).

−4·2+3a=7

Exercise Set 8.4

2. Let x= the number of small stems sold and y= the num-

ber of large stems sold. Solve:

4. Let xand yrepresent the number of truﬄes and cream

mints, respectively, in each box. The 75 boxes are identi-

6. Let x= the number of ounces of lemon juice and y= the

number of ounces of linseed oil to be used. Solve:

8. Let x= the number of pounds of Deep Thought Granola

and y= the number of pounds of Oat Dream Granola to

be used. Solve:

10. Let x= the number of pounds of soybean meal and y=

the number of pounds of corn meal in the mixture. The

0.16x+0.09y=42

12. Let xand yrepresent the number of Acuminata bulbs and

Cafe Brun bulbs, respectively, in the assortment. Solve:

212 Chapter 8: Systems of Equations

14. Let x= the amount invested at 4% and y= the amount

invested at 6%. Solve:

16. Let x= the number of liters of Arctic Antifreeze that

should be used and let y= the number of liters in the

x+y=31,000,

0.028x+0.045y= 1024.40

20. Let xand yrepresent the number of quarters and dollar

Distance Rate Time

Down-

26. Let w= the speed of the headwind. Note that 4 hr 50

min=4

5

6hr, or 29

6hr.

Distance Rate Time

Solve: 2900 = (r−w)5,

Slower

d190 t

30. Distance Rate Time

Toward

Solve: d= 370t,

34. f(−2.5) = 4(−2.5) −7=−10 −7=−17

40. Let b= the number of boys and g= the number of girls.

Chapter 8 Mid-Chapter Review

y

4

5

x

+

y

= 3

x

1—1—33254—2—4—5

—1

—2

1

2

4

x

– 2

y

= 6

y

2

3

4

5

2

y

– 2

x

= 6

4. True; a vertical line x=aand a horizontal line y=b

intersect at exactly one point, (a, b).

5. x+2y=3,

6. 3x−2y=5,

6+4y=14

7. Graph the lines on the same set of axes.

The solution appears to be (5,−1).

Check:

y=x−6y=4−x

8. Graph the lines on the same set of axes.

x+y=3 3x+y=3

0+3 ? 3 3·0+3 ? 3

9. Graph the lines on the same set of axes.

10. Graph the lines on the same set of axes.

214 Chapter 8: Systems of Equations

11. x=y+2,(1)

x=6+2=8

12. y=x−5,(1)

x−2y= 8 (2)

Substitute x−5 for yin the Equation (2) and solve for x.

x=2

13. 4x+3y=3,(1)

14. 3x−2y=1,(1)

x=y+ 1 (2)

15. 2x+y=2,(1)

2x= 10 Adding

x=5

Substitute 5 for xin Equation (2) and solve for y.

17. 3x−4y=5,(1)

3x−4y=5

2x+3y= 9 (2)

We use the multiplication principle on both equations and

then add.

19. x−2y=5,(1)

20. 4x−6y=2,(1)

−2x+3y=−1 (2)

21. 1

2x+1

3y=1,(1)

x=10

Substitute 10 for xin Equation (3) and solve for y.

22. 0.2x+0.3y=0.6,(1)

Substitute 8 for yin Equation (2) and solve for x.

Translate.

Perimeter is 44 ft.

Solve. We substitute l−2 for win Equation (1) and solve

for l.

2l+2(l−2)=44

24. Familiarize. Let xand yrepresent the amounts invested

Interest earned

  was $129.



216 Chapter 8: Systems of Equations

−2x−2y=−10,000

2x+3y=12,900

25. Familiarize. Let x= the number of liters of 20% solution

and y= the number of liters of 50% solution to be used.

The mixture contains 30% (84 L), or 0.3(84 L), or 25.2 L

Amount

of acid 0.2x0.5y25.2 L

then add.

−2x−2y=−168

0.5(28 L) = 11.2L+14L=25.2 L. The answer checks.

26. Familiarize. Let d= the distance traveled and r= the

speed of the boat in still water, in mph. Then when the

for r.

8r−48 = 5r+30

3r−48 = 30

27. Graphically: 1. Graph y=3

4x+ 2 and y=2

5x−5 and

28. a) Answers may vary.

x+y=1,

29. Answers may vary. Form a linear expression in two vari-

Exercise Set 8.5 217

30. Answers may vary. Let any linear equation be one equation

Exercise Set 8.5

2. 2x−y−4z=−12,(1)

16x−12z=−44 Multiplying (4) by 4

4. x−y+z=4,(1)

3x−2y+3z= 7 (2)

1−y+ 2 = 4 Substituting in (1)

6. 6x−4y+5z=31,(1)

5x+2y+2z=13,(2)

x+y+z= 2 (3)

8. x+y+z=0,(1)

z=−4

10. 2x+y−3z=−4,(1)

218 Chapter 8: Systems of Equations

x=2

12. 2x+y+2z=11,(1)

3x+2y+2z= 8 (2)

−3x−12y−9z= 0 Multiplying (3) by −3

z=6

14. 2x+y+2z=3,(1)

16. 2r+s+t=6,(1)

2r+s+t= 6 (1)

t=3

r+4·3 = 16 Substituting in (5)

r+12=16

18. x+4y−z=5,(1)

2x−y+3z=−5 (2)

5x+11y= 10 (4)

Exercise Set 8.5 219

20. 10x+6y+z=7,(1)

22. 3p+2r=11,(1)

6q−42r= 24 Multiplying (2) by 6

24. 4a+9b=8,(1)

c=−5

26. a−5c=17,(1)

28. F=3ab

30. F=1

2t(c−d)

2F=tc −td

tc −2F

t=d, or

34. y=−2

3x−5

4

The equation is in slope-intercept form, y=mx +b. The

36. 2x−5y=10

y=2

5x−2

38. w+x−y+z=0,(1)

2w−x−y+3z= 7 (4)

Pairing equations (1) and (2), (2) and (3), and (2) and (4)

Substituting −1 for xin (5) and (7), we have

2w−3y=−6,(8)

Exercise Set 8.6

2. Let x,y, and zrepresent the fat content of a Big Mac,

fat.

4. Let x,y, and zrepresent the measures of angles A,B, and

x+y+z=180,

y=2x,

6. We know the thousands digit must be 1. The United States

We let xrepresent the hundreds digit, ythe tens digit, and

We also know that zis one more than x.

When z=3,x=2.

8. Let x,y, and zrepresent the number of tall, grande and

venti coﬀees served, respectively. Solve:

Exercise Set 8.6 221

10. Let x,y, and zrepresent the prices of a children’s book, a

paperback, and a hardback, respectively. Solve:

12. Let x,y, and zrepresent the prices of upgrading the pro-

cessor, the memory, and the graphics card, respectively.

Solve:

14. Let x,y, and zrepresent the number of calories in the

sandwich, the soup, and the cookie respectively. Solve:

16. Let x,y, and zrepresent the number of servings of roast

beef, baked potato, and asparagus, respectively. Solve:

18. Let x,y, and zrepresent the amount of the Perkins loan,

the Staﬀord loan, and the bank loan, respectively. Solve:

20. Let x,y, and zrepresent the number of 2-point field goals,

22. Let x,y, and zrepresent the number of orders that can be

processed alone by Steven, Teri, and Isaiah, respectively.

24.

26. It is possible for a vertical line to intersect the graph more

tickets to Tom and Gary, each has the following number

of tickets:

Tom: T+T, or 2T,

2(H−T−G),

Tom: 2T−2G−(H−T−G),or

2(3T−H−G),