Chapter 4

ⓐ yes ⓑ no

ⓐ no ⓑ yes

$\left(3,2\right)$

$\left(2,3\right)$

$\left(3,4\right)$

$\left(5,\mathrm{-4}\right)$

no solution

no solution

infinitely many solutions

infinitely many solutions

ⓐ no solution, inconsistent, independent ⓑ one solution, consistent, independent

ⓐ no solution, inconsistent, independent ⓑ one solution, consistent, independent

$\left(6,1\right)$

$\left(\mathrm{-3},5\right)$

$\left(2,\frac{3}{2}\right)$

$\left(-\frac{1}{2},\mathrm{-2}\right)$

$\left(2,\mathrm{-1}\right)$

$\left(\mathrm{-2},3\right)$

$\left(1,3\right)$

$\left(4,\mathrm{-3}\right)$

$\left(6,2\right)$

$\left(1,\mathrm{-2}\right)$

infinitely many solutions

infinitely many solutions

ⓐ Since both equations are in standard form, using elimination will be most convenient. ⓑ Since one equation is already solved for x, using substitution will be most convenient.

ⓐ Since one equation is already solved for y, using substitution will be most convenient. ⓑ Since both equations are in standard form, using elimination will be most convenient.

3, 7

2, $\mathrm{-8}$

160 policies

1000 suits

Mark burned 11 calories for each minute of yoga and 7 calories for each minute of jumping jacks.

Erin burned 11 calories for each minute on the rowing machine and 5 calories for each minute of weight lifting.

The angle measures are 55 and 35.

The angle measures are 5 and 85.

The angle measures are 42 and 138.

The angle measures are 66 and 114.

22, 68

36, 54

The length is 60 feet and the width is 35 feet.

The length is 60 feet and the width is 38 feet.

It will take Clark 4 hours to catch Mitchell.

It will take Sally $1\frac{1}{2}$ hours to catch up to Charlie.

The rate of the boat is 11 mph and the rate of the current is 1 mph.

The speed of the canoe is 7 mph and the speed of the current is 1 mph.

The speed of the jet is 235 mph and the speed of the wind is 30 mph.

The speed of the jet is 408 mph and the speed of the wind is 24 mph.

206 adults, 347 children

42 adults, 105 children

13 dimes and 29 quarters

19 quarters and 51 nickels

3 pounds peanuts and 2 pounds cashews

10 pounds of beans, 10 pounds of ground beef

120 ml of 25% solution and 30 ml of 50% solution

125 ml of 10% solution and 125 ml of 40% solution

$42,000 in the stock fund and $8000 in the savings account

$1750 at 11% and $5250 at 13%

Bank $4,000; Federal $14,000

$41,200 at 4.5%, $24,000 at 7.2%

ⓐ $C(x)=15x+25,500$

ⓑ $R(x)=32x$

ⓒ

ⓓ $\mathrm{1,500}$; when 1,500 benches are sold, the cost and revenue will be both 48,000

ⓐ $C(x)=120x+\mathrm{150,000}$

ⓑ $R(x)=170x$

ⓒ

ⓓ $\mathrm{3,000}$; when 3,000 benches are sold, the revenue and costs are both $510,000

ⓐ yes ⓑ no

ⓐ no ⓑ yes

$\left(2,\mathrm{-1},3\right)$

$\left(\mathrm{-2},3,4\right)$

$\left(\mathrm{-3},4,\mathrm{-2}\right)$

$\left(\mathrm{-2},3,\mathrm{-1}\right)$

no solution

no solution

infinitely many solutions$\left(x,3,z\right)$ where $x=z-3;y=3;z$ is any real number

infinitely many solutions $\left(x,y,z\right)$ where$x=5z-2;y=4z-3;z$ is any real number

The fine arts department sold 75 adult tickets, 200 student tickets, and 75 child tickets.

The soccer team sold 200 adult tickets, 300 student tickets, and 100 child tickets.

ⓐ $\left[\begin{array}{ccccccc}3\hfill & & & 8\hfill & & & \hfill \mathrm{-3}\\ 2\hfill & & & 5\hfill & & & \hfill \mathrm{-3}\end{array}\right]$
ⓑ $\left[\begin{array}{cccccccccc}2\hfill & & & \hfill \mathrm{-5}& & & 3\hfill & & & \hfill 8\\ 3\hfill & & & \hfill \mathrm{-1}& & & 4\hfill & & & \hfill 7\\ 1\hfill & & & \hfill 3& & & 2\hfill & & & \hfill \mathrm{-3}\end{array}\right]$

ⓐ $\left[\begin{array}{ccccccc}11\hfill & & & 9\hfill & & & \hfill \mathrm{-5}\\ 7\hfill & & & 5\hfill & & & \hfill \mathrm{-1}\end{array}\right]$
ⓑ $\left[\begin{array}{cccccccccc}5\hfill & & & \hfill \mathrm{-3}& & & \hfill 2& & & \hfill \mathrm{-5}\\ 2\hfill & & & \hfill \mathrm{-1}& & & \hfill \mathrm{-1}& & & \hfill 4\\ 3\hfill & & & \hfill \mathrm{-2}& & & \hfill 2& & & \hfill \mathrm{-7}\end{array}\right]$

$\{\begin{array}{c}\text{<mspace width="0.2em"> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>=</mo> <mn>3</mn> </mspace>}\hfill \\ 2x+y-2z=1\hfill \\ \text{<mspace width="0.2em"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mspace>}\hfill \end{array}$

$\{\begin{array}{c}\text{<mspace width="0.2em"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>4</mn> </mspace>}\hfill \\ 2x+3y-z=8\hfill \\ x+y-z=3\hfill \end{array}$

ⓐ $\left[\begin{array}{cccccccccc}\hfill \mathrm{-2}& & & \hfill 3& & & \hfill 0& & & \hfill \mathrm{-2}\\ \hfill 4& & & \hfill \mathrm{-1}& & & \hfill \mathrm{-4}& & & \hfill 4\\ \hfill 5& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-2}\end{array}\right]$
ⓑ $\left[\begin{array}{cccccccccc}\hfill \mathrm{-2}& & & \hfill 3& & & \hfill 0& & & \hfill \mathrm{-2}\\ \hfill 4& & & \hfill \mathrm{-1}& & & \hfill \mathrm{-4}& & & \hfill 4\\ 15\hfill & & & \hfill \mathrm{-6}& & & \hfill \mathrm{-6}& & & \hfill \mathrm{-6}\end{array}\right]$
ⓒ $\left[\begin{array}{ccccccccccccc}\hfill \mathrm{-2}& & & \hfill 3& & & \hfill 0& & & \hfill \mathrm{-2}& & & \\ \hfill 3& & & \hfill 4& & & \hfill \mathrm{-13}& & & \hfill \mathrm{-16}& & & \hfill \mathrm{-8}\\ \hfill 15& & & \hfill \mathrm{-6}& & & \hfill \mathrm{-6}& & & \hfill \mathrm{-6}& & & \end{array}\right]$

ⓐ $\left[\begin{array}{cccccccccc}4\hfill & & & \hfill 1& & & \hfill \mathrm{-3}& & & \hfill 2\\ 2\hfill & & & \hfill \mathrm{-3}& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-4}\\ 5\hfill & & & \hfill 0& & & \hfill 4& & & \hfill \mathrm{-1}\end{array}\right]$
ⓑ $\left[\begin{array}{cccccccccc}8\hfill & & & \hfill 2& & & \hfill \mathrm{-6}& & & \hfill 4\\ 2\hfill & & & \hfill \mathrm{-3}& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-4}\\ 5\hfill & & & \hfill 0& & & \hfill 4& & & \hfill \mathrm{-1}\end{array}\right]$
ⓒ $\left[\begin{array}{cccccccccc}14\hfill & & & \hfill \mathrm{-7}& & & \hfill \mathrm{-12}& & & \hfill \mathrm{-8}\\ 2\hfill & & & \hfill \mathrm{-3}& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-4}\\ 5\hfill & & & \hfill 0& & & \hfill 4& & & \hfill \mathrm{-1}\end{array}\right]$

$\left[\begin{array}{ccccccc}1\hfill & & & \hfill \mathrm{-1}& & & \hfill 2\\ 0\hfill & & & \hfill \mathrm{-3}& & & \hfill \mathrm{-4}\end{array}\right]$

$\left[\begin{array}{ccccccc}1\hfill & & & \hfill \mathrm{-1}& & & \hfill 3\\ 0\hfill & & & \hfill \mathrm{-5}& & & \hfill 8\end{array}\right]$

The solution is $\left(4,\mathrm{-1}\right).$

The solution is $\left(\mathrm{-2},0\right).$

$\left(6,\mathrm{-1},\mathrm{-3}\right)$

$(5,7,4)$

no solution

no solution

infinitely many solutions $\left(x,y,z\right)\text{,}$ where $x=z-3;y=3;z$ is any real number.

infinitely many solutions $\left(x,y,z\right)\text{,}$ where $x=5z-2;y=4z-3;z$ is any real number.

ⓐ $\mathrm{-14};$ ⓑ $\mathrm{-28}$

ⓐ 2 ⓑ $\mathrm{-15}$

ⓐ 3 ⓑ 11 ⓒ 2

ⓐ $\mathrm{-3}$ ⓑ 2 ⓒ 3

37

7

$\mathrm{-11}$

8

$(-\frac{15}{7},\frac{24}{7})$

$(\mathrm{-2},0)$

$\left(\mathrm{-9},3,\mathrm{-1}\right)$

$\left(\mathrm{-6},3,\mathrm{-2}\right)$

no solution

infinite solutions

yes

yes

ⓐ no ⓑ yes

ⓐ yes ⓑ no

ⓐ $\{\begin{array}{c}30m+20p\le 160\hfill \\ 2m+3p\le 15\hfill \end{array}$
ⓑ

ⓒ yes
ⓓ no

ⓐ $\{\begin{array}{c}a\ge p+5\hfill \\ a+2p\le 400\hfill \end{array}$
ⓑ

ⓒ no
ⓓ no

ⓐ $\{\begin{array}{c}0.75d+2e\le 25\hfill \\ 360d+110e\ge 1000\hfill \end{array}$
ⓑ

ⓒ yes
ⓓ no

ⓐ $\{\begin{array}{c}140p+125j\ge 1000\hfill \\ 1.80p+1.25j\le 12\hfill \end{array}$
ⓑ

ⓒ yes
ⓓ no

ⓐ yes ⓑ no

ⓐ yes ⓑ no

$(\mathrm{-2},3)$

$(0,2)$

$(2,4)$

$(\mathrm{-2},2)$

$(3,3)$

$(6,\mathrm{-4})$

no solution

no solution

infinite solutions

infinite solutions

No solutions, inconsistent, independent

1 point, consistent and independent

infinite solutions, consistent, dependent

$\left(1,\mathrm{-4}\right)$

$\left(\mathrm{-3},2\right)$

$\left(\mathrm{-1}\text{/}2,5\text{/}2\right)$

$\left(\mathrm{-5},4\right)$

$\left(0,10\right)$

$\left(4,\mathrm{-2}\right)$

$\left(4,0\right)$

none

$\left(4,5\right)$

$\left(7,12\right)$

$\left(\mathrm{-3},\mathrm{-5}\right)$

$\left(2,\mathrm{-3}\right)$

$\left(\mathrm{-11},2\right)$

$\left(6\text{/}\mathrm{-9},24\text{/}7\right)$

$\left(\mathrm{-3},2\right)$

infinitely many

infinitely many

ⓐ substitution ⓑ elimination

ⓐ elimination ⓑ substituion

Answers will vary.

Answers will vary.

13 and 17

$\mathrm{-7}$ and $\mathrm{-19}$

14 and 23

22 and $\mathrm{-67}$

Eighty cable packages would need to be sold to make the total pay the same.

Mitchell would need to sell 120 stoves for the companies to be equal.

8 and 40 gallons

1000 calories playing basketball and 400 calories canoeing

Oranges cost $2 per pound and bananas cost $1 per pound

Package of paper $4, stapler $7

Hot dog 150 calories, cup of cottage cheese 220 calories

Owen will need 80 quarts of water and 20 quarts of concentrate to make 100 quarts of lemonade.

$53.5$ degrees and $36.5$ degrees

16 degrees and 74 degrees

134 degrees and 46 degrees

37 degrees and 143 degrees

$16\text{°}$ and $74\text{°}$

$45\text{°}$ and $45\text{°}$

Width is 41 feet and length is 118 feet.

Width is 10 feet and length is 40 feet.

11 hours

$1.5$ hour

Boat rate is 16 mph and current rate is 4 mph.

Boat rate is 18 mph and current rate is 2 mph.

Jet rate is 265 mph and wind speed is 22 mph.

Jet rate is 415 mph and wind speed is 25 mph.

Answers will vary.

110 adult tickets, 190 child tickets

6 good seats, 10 cheap seats

92 adult tickets, 220 children tickets

13 nickels, 3 dimes

42 dimes, 8 quarters

17 $10 bills, 37 $20 bills

80 pounds nuts and 40 pounds raisins

9 pounds of Chicory coffee, 3 pounds of Jamaican Blue Mountain coffee

10 bags of M&M’s, 15 bags of Reese’s Pieces

$7.5$ liters of each solution

80 liters of the 25% solution and 40 liters of the 10% solution

240 liters of the 90% solution and 120 liters of the 75% solution

$1600 at 8%, 960 at 6%

$28,000 at 9%, $36,000 at $5.5\text{\%}$

$8500 CD, $1500 savings account

$55,000 on loan at 6% and $30,000 on loan at $4.5\text{\%}$

ⓐ $C(x)=5x+6500$

ⓑ $R(x)=10x$

ⓒ

ⓓ 1,500; when 1,500 water bottles are sold, the cost and the revenue equal $15,000

Answers will vary.

ⓐ no ⓑ yes

ⓐ no ⓑ yes

$\left(4,5,2\right)$

$\left(7,12,\mathrm{-2}\right)$

$\left(\mathrm{-3},\mathrm{-5},4\right)$

$\left(2,\mathrm{-3},\mathrm{-2}\right)$

$\left(6,\mathrm{-9},\mathrm{-3}\right)$

$\left(3,\mathrm{-4},\mathrm{-2}\right)$

$\left(\mathrm{-3},2,3\right)$

$\left(\mathrm{-2},0,\mathrm{-3}\right)$

no solution

$x=\frac{203}{16};y=\frac{\mathrm{–25}}{16};z=\frac{\mathrm{–231}}{16};$

$\left(x,y,z\right)$ where $x=5z+2;y=\mathrm{-3}z+1;z$ is any real number

$\left(x,y,z\right)$ where $x=5z-2;y=4z-3;z$ is any real number

42, 50, 58

$20, $5, $10

Answers will vary.

ⓐ $\left[\begin{array}{ccccccc}2\hfill & & & \hfill 4& & & \hfill \mathrm{-5}\\ 3\hfill & & & \hfill \mathrm{-2}& & & \hfill 2\end{array}\right]$
ⓑ $\left[\begin{array}{cccccccccc}\hfill 3& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-1}& & & \hfill \mathrm{-2}\\ \hfill \mathrm{-2}& & & \hfill 1& & & \hfill 0& & & \hfill 5\\ \hfill 5& & & \hfill 4& & & \hfill 1& & & \hfill \mathrm{-1}\end{array}\right]$

ⓐ $\left[\begin{array}{ccccccc}2\hfill & & & \hfill \mathrm{-5}& & & \hfill \mathrm{-3}\\ 4\hfill & & & \hfill \mathrm{-3}& & & \hfill \mathrm{-1}\end{array}\right]$
ⓑ $\left[\begin{array}{cccccccccc}4\hfill & & & \hfill 3& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-3}\\ \hfill \mathrm{-2}& & & \hfill 1& & & \hfill \mathrm{-3}& & & \hfill 4\\ \mathrm{-1}\hfill & & & \hfill \mathrm{-4}& & & \hfill 5& & & \hfill \mathrm{-2}\end{array}\right]$

$\{\begin{array}{c}2x-4y=\mathrm{-2}\hfill \\ 3x-3y=\mathrm{-1}\hfill \end{array}$

$\{\begin{array}{c}2x-2y=\mathrm{-1}\hfill \\ 2y-z=2\hfill \\ 3x-z=\mathrm{-2}\hfill \end{array}$

ⓐ $\left[\begin{array}{ccccccc}3\hfill & & & \hfill 2& & & \hfill 1\\ 4\hfill & & & \hfill \mathrm{-6}& & & \hfill \mathrm{-3}\end{array}\right]$
ⓑ $\left[\begin{array}{ccccccc}12\hfill & & & \hfill 8& & & \hfill 4\\ 4\hfill & & & \hfill \mathrm{-6}& & & \hfill \mathrm{-3}\end{array}\right]$
ⓒ $\left[\begin{array}{ccccccc}12\hfill & & & \hfill 8& & & \hfill 4\\ 24\hfill & & & \hfill \mathrm{-10}& & & \hfill \mathrm{-5}\end{array}\right]$

ⓐ $\left[\begin{array}{cccccccccc}\hfill 2& & & \hfill 1& & & \hfill \mathrm{-4}& & & \hfill 5\\ \hfill 6& & & \hfill \mathrm{-5}& & & \hfill 2& & & \hfill 3\\ \hfill 3& & & \hfill \mathrm{-3}& & & \hfill 1& & & \hfill \mathrm{-1}\end{array}\right]$
ⓑ $\left[\begin{array}{cccccccccc}2\hfill & & & \hfill 1& & & \hfill \mathrm{-4}& & & \hfill 5\\ 6\hfill & & & \hfill \mathrm{-5}& & & \hfill 2& & & \hfill 3\\ 3\hfill & & & \hfill \mathrm{-3}& & & \hfill 1& & & \hfill \mathrm{-1}\end{array}\right]$
ⓒ $\left[\begin{array}{cccccccccc}\hfill 2& & & \hfill 1& & & \hfill \mathrm{-4}& & & \hfill 5\\ \hfill 6& & & \hfill \mathrm{-5}& & & \hfill 2& & & \hfill 3\\ \hfill \mathrm{-4}& & & \hfill 7& & & \hfill \mathrm{-6}& & & \hfill 7\end{array}\right]$

$\left[\begin{array}{cccccccccc}1\hfill & & & \hfill \mathrm{-2}& & & \hfill 3& & & \hfill \mathrm{-4}\\ 0\hfill & & & \hfill 5& & & \hfill \mathrm{-11}& & & \hfill 17\\ 0\hfill & & & \hfill 1& & & \hfill \mathrm{-10}& & & \hfill 7\end{array}\right]$

$\left(1,\mathrm{-1}\right)$

$\left(3,3\right)$

$(\mathrm{-2},5,2)$

$\left(\mathrm{-3},\mathrm{-5},4\right)$

$\left(\mathrm{-3},2,3\right)$

$\left(\mathrm{-2},0,\mathrm{-3}\right)$

no solution

no solution

infinitely many solutions $\left(x,y,z\right)$ where $x=\frac{1}{2}z+4;y=\frac{1}{2}z-6;z$ is any real number

infinitely many solutions $\left(x,y,z\right)$ where $x=5z+2;y=\mathrm{-3}z+1;z$ is any real number

Answers will vary.

4

10

ⓐ 6 ⓑ $\mathrm{-14}$ ⓒ $\mathrm{-6}$

ⓐ 9 ⓑ $\mathrm{-3}$ ⓒ 8

$\mathrm{-77}$

49

$\mathrm{-24}$

25

$(7,6)$

$(\mathrm{-2},0)$

$(\mathrm{-3},2)$

$(\mathrm{-9},3)$

$\left(\mathrm{-3},\mathrm{-5},4\right)$

$\left(2,\mathrm{-3},\mathrm{-2}\right)$

$\left(\mathrm{-3},2,3\right)$

$\left(\mathrm{-2},0,\mathrm{-3}\right)$

infinitely many solutions

inconsistent

inconsistent

infinitely many solutions

yes

no

Answers will vary.

Answers will vary.

ⓐ false ⓑ true

ⓐ false ⓑ true

ⓐ false ⓑ true

ⓐ $\{\begin{array}{c}f\ge 0\hfill \\ p\ge 0\hfill \\ f+p\le 20\hfill \\ 2f+5p\le 50\hfill \end{array}$
ⓑ

ⓒ yes
ⓓ no

ⓐ $\{\begin{array}{c}c\ge 0\hfill \\ a\ge 0\hfill \\ c+a\le 24\hfill \\ a\ge 3c\hfill \end{array}$
ⓑ

ⓒ yes
ⓓ no

ⓐ $\{\begin{array}{c}w\ge 0\hfill \\ b\ge 0\hfill \\ 27w+16b>80\hfill \\ 3.20w+1.75b\le 10\hfill \end{array}$
ⓑ

ⓒ no
ⓓ yes

ⓐ $\{\begin{array}{c}w\ge 0\hfill \\ r\ge 0\hfill \\ w+r\ge 4\hfill \\ 270w+650r\ge 1500\hfill \end{array}$
ⓑ

ⓒ no
ⓓ yes

Answers will vary.

ⓐ yes ⓑ no

one solution, consistent system, independent equations

$\left(4,5\right)$

$\left(3,1\right)$

infinitely many solutions

$\left(4,\mathrm{-1}\right)$

$\left(6,2\right)$

elimination

50 irises and 150 tulips

10 calories jogging and 10 calories cycling

119, 61

$35\text{°}$ and $55\text{°}$

the length is 450 feet, the width is 264 feet

$\frac{1}{2}$ an hour

the rate of the jet is 395 mph, the rate of the wind is 7 mph

41 dimes and 11 pennies

$46\frac{2}{3}$ liters of 30% solution, $23\frac{1}{3}$ liters of 60% solution

$29,000 for the federal loan, $14,000 for the private loan

ⓐ no ⓑ yes

$\left(\mathrm{-3},2,\mathrm{-4}\right)$

no solution

25, 20, 15

$\left[\begin{array}{cccccccccc}\hfill 4& & & \hfill 3& & & \hfill 0& & & \hfill \mathrm{-2}\\ \hfill 1& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-3}& & & \hfill 7\\ \hfill 2& & & \hfill \mathrm{-1}& & & \hfill 2& & & \hfill \mathrm{-6}\end{array}\right]$