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Solve a system of equations using any method

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Darrell's Bakery recently spent a total of

\$115

on new equipment, and their average hourly operating costs are

\$15

. Their average hourly receipts are

\$16

. The bakery will soon make back the amount it invested in equipment. What would the total expenses and receipts both equal? How many hours will that take?

\newline

The bakery's expenses and receipts will both total

\$

_____ after _____ hours.

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Brenda and Eduardo each want to run for president of their school's student body council. In order to do so, they must collect a certain number of signatures and get a nomination. So far, Brenda has

20

signatures, and Eduardo has

28

. Brenda is collecting signatures at an average rate of

9

per day, whereas Eduardo is averaging

8

signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How long will that take? How many signatures will they both have?

\newline

\_\_

days, Brenda and Eduardo will each have collected

\_\_

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

The volleyball team and the wrestling team at Westminster High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets

\$4

per car. In addition, they have already brought in

\$73

from past fundraisers. The wrestling team has raised

\$31

in the past, and they are making

\$5

per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take?

\newline

Each team will have raised a total of

\$\_\_\_\_\_

\_\_\_\_\_

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

A couple of friends decide to race each other. Mia can run

5 \text{ meters per second}

, whereas Neil can run

8 \text{ meters per second}

. Because she is slower, Mia also gets a head start of

36 \text{ meters}

. Shortly after they start running, Neil will catch up to Mia. How long will that take? How far will Neil have to run?

\newline

\_\_

seconds for Neil to run

\_\_

meters and catch up to Mia.

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Vicky works in the shipping department of a toy manufacturer. Toy cars weigh

3

pounds apiece and are shipped in a container that weighs

12

pounds when empty. Toy trucks, which weigh

8

pounds apiece, are shipped in a container weighing

7

pounds. When packed with toys and ready for shipment, both kinds of containers have the same number of toys and the same weight. What is the weight of each container? What is the number of toys?

\newline

Each container weighs

\underline{\hspace{2em}}

pounds and contains

\underline{\hspace{2em}}

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

The Harding family is looking to rent a large truck for their upcoming move. With Billy's Moving, they would pay

\$40

for the first day plus

\$7

per additional day. With Wildgrove Rent-a-Truck, in comparison, the family would pay

\$41

for the first day plus

\$6

per additional day. Before deciding on which company to use, Mrs. Harding wants to find out what number of additional days would make the two choices equivalent with regards to cost. What would the total cost be? How many additional days would that be?

\newline

The Harding family would pay

\$\_\_\_\_

either way if they rented the truck for

\_\_\_

additional days.

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

The members of a sewing circle are making blankets to give to shelters. This week, they made

44

twin-size blankets and

11

queen-size blankets, using a total of

165

meters of fabric. Last week, the members completed

16

twin-size blankets and

32

queen-size blankets, which required

256

total meters of fabric. How much fabric is used for the different sizes of blankets?

\newline

A twin-size blanket uses

\_

meters of fabric and a queen-size one uses

\_

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Clarence and Angie went to an arcade where the machines took tokens. Clarence played

6

games of skee ball and

2

games of pinball, using a total of

12

tokens. At the same time, Angie played

6

games of skee ball and

5

games of pinball, using up

21

tokens. How many tokens does each game require?

\newline

Every game of skee ball requires

\_\_

tokens, and every game of pinball requires

\_\_

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Lillian went to play miniature golf on Monday, when it cost

\$12

to rent the club and ball, plus

\$2

per game. Janelle went Thursday, paying

\$5

per game, plus rental fees of

\$3

. By coincidence, they played the same number of games for the same total cost. How many games did each one play? How much did each one spend?

\newline

Lillian and Janelle each played

\underline{\quad}

games and spent

\$\underline{\quad}

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Bryan is a server at an all-you-can eat sushi restaurant. At one table, the customers ordered

2

child buffets and

3

adult buffets, which cost a total of

\$121

. At another table, the customers ordered

3

child buffets and

3

adult buffets, paying a total of

\$138

. How much does the buffet cost for each child and adult?

\newline

The cost for a child is

\$

_____, and the cost for an adult is

\$

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Madelyn is going to ship some gifts to family members, and she is considering two shipping companies. The first shipping company charges a fee of

\$12

to ship a medium box, plus an additional

\$3

per pound. A second shipping company charges

\$15

for the same size of box, plus an additional

\$2

per pound. At a certain weight, the two shipping methods will cost the same amount. What is that weight? How much will it cost?

\newline

\underline{\hspace{2em}}

pounds, the two shipping methods both cost

\$\underline{\hspace{2em}}

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

When Colin does

17

13

sit-ups, it takes a total of

64

seconds. In comparison, he needs

80

20

20

sit-ups. How long does it take Colin to do each kind of exercise?

\newline

\_\_\_

seconds to do a push-up and

\_\_\_

seconds to do a sit-up.

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Dylan and Leo are hosting events that are catered by the same company. Dylan plans to have

87

96

children attend, so the total projected cost of his meals is

\$4,938

70

96

children on his guest list, so he will pay the caterer

\$4,292

. How much does the caterer charge for each meal?

\newline

Every adult's meal costs

\$

_____, and every child's meal costs

\$

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

A camp counselor for a summer day camp sometimes buys lunch for her campers at a nearby fast food restaurant. On Monday, she purchased

1

hamburger kid meal and

3

chicken nugget kid meals, for a total of

\$14

. On Thursday, she spent

\$36

4

hamburger kid meals and

7

chicken nugget kid meals. How much does each type of meal cost?

\newline

Each hamburger meal costs

\$

_____, and each chicken nugget meal costs

\$

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Janelle and Duncan are comparing the international calling plans on their cell phones. On her plan, Janelle pays

\$1

just to place a call and

\$5

for each minute. When Duncan makes an international call, he pays

\$5

to place the call and

\$1

for each minute. A call of a certain duration would cost exactly the same under both plans. What is the duration? What is the cost?

\newline

\_\_

minutes would cost

\$\_\_

under each plan.

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

A realtor is decorating some homes for sale, putting a certain number of decorative pillows on each twin bed and a certain number on each queen bed. In one house, she decorated

4

2

queen beds and used a total of

42

pillows. At another house, she used

31

pillows to spruce up

4

1

queen bed. How many decorative pillows did the realtor arrange on each bed?

\newline

The realtor used

\_\_\_

pillows on every twin bed and

\_\_\_

pillows on every queen bed.

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Nick and his cousin are playing a game where they pick up colored sticks. Nick currently has

14

points and likes to pick up the green sticks, earning

9

points every turn. His cousin just lost all her points on the previous turn, and has a strategy to catch up by getting all the pink ones, earning

10

points per turn. In a certain number of turns, the score will be tied. How long will that take? How many points will they each have?

\newline

\_

turns, both players will have

\_

r

\newline

-10 - 5r = -6r - 1

r

-10 - 5r = -6r - 1

Solve each equation.

\newline

3x-18=0

x

\newline

3x+8=-1

\newline

x=\square

Natasha and her two dogs were walking on a perfectly straight road when her two dogs ran away from her in opposite directions. Her beagle is now

\frac{25}{4}

meters directly to her right, and her labrador is

\frac{51}{20}

meters directly to her left.

\newline

2

of the following expressions represents how far apart the two dogs are?

\newline

2

\newline

\left|\frac{25}{4}+\frac{51}{20}\right|

\newline

\left|-\frac{51}{20}+\frac{25}{4}\right|

\newline

\left|-\frac{51}{20}-\frac{25}{4}\right|

Becky tried to evaluate an expression step by step.

\newline

\begin{array}{l} \frac{4}{5}+7-\frac{5}{4} \\ =\frac{4}{5}-\frac{5}{4}+7 \quad \text { Step } 1 \\ =0+7 \quad \text { Step 2 } \\ =7 \quad \text { Step } 3 \\ \end{array}

\newline

Find Becky's first mistake.

\newline

1

\newline

(A) Commuting a negative term changes the value.

\newline

\frac{4}{5}

\frac{5}{4}

-\frac{9}{20}

0

\newline

0

7

0

7

Raul tried to evaluate an expression step by step.

\newline

\begin{array}{l} 3-(-5)-7 \\ =3+5-7 \quad \text { Step } 1 \\ =3+2 \quad \text { Step 2 } \\ =5 \quad \text { Step } 3 \\ \end{array}

\newline

Find Raul's first mistake.

\newline

1

\newline

(A) The opposite of

-5

-5

5

\newline

5

7

-2

2

\newline

3

2

6

5

Amber tried to evaluate an expression step by step.

\newline

\begin{array}{l} -3-(-5)+4 \\ =-3+5+4 \quad \text { Step } 1 \\ =-3+9 \quad \text { Step 2 } \\ =-12 \quad \text { Step } 3 \\ \end{array}

\newline

Find Amber's first mistake.

\newline

1

\newline

(A) The opposite of

-5

-5

5

\newline

(B) Adding from right to left changes the sum's value.

\newline

-3

9

6

-12

New policies reduced the overuse of water in an area. The water table in that area went up by

1

2

meters in one year. The water table was

12

5

meters below the surface at the start of the year.

\newline

The following equation describes this situation.

\newline

-12.5+1.2=-11.3

\newline

-11

3

\newline

1

\newline

(A) The water table was

11

3

meters below the surface at the end of the year.

\newline

(B) The water table went down by

11

3

meters in one year.

\newline

(C) Each household used

11

3

fewer liters of water during the year.

Josiah is cooking a piece of lamb. Before he puts it in the oven, the internal temperature of the lamb is

41^{\circ} \mathrm{C}

below the final temperature it needs to reach. The outside of the lamb heats up faster than the inside. Josiah takes the lamb out of the oven when the internal temperature is still

6^{\circ} \mathrm{C}

below the final temperature it needs to reach, so the inside and outside can balance out to the right final temperature.

\newline

Josiah wonders how much the internal temperature had changed when he removed the lamb from the oven.

\newline

Which of the following equations matches the situation above?

\newline

1

\newline

41+6=

\newline

-41+?=-6

\newline

41+?=6

Eric was rock climbing. At one point, he stopped and climbed straight down

2 \frac{1}{2}

meters. Then he climbed straight up

6 \frac{3}{4}

meters. Eric was wondering what his change in elevation was after these two moves.

\newline

Which of the following equations matches the situation above?

\newline

1

\newline

-2 \frac{1}{2}-6 \frac{3}{4}=

\newline

2 \frac{1}{2}-6 \frac{3}{4}=

\newline

-2 \frac{1}{2}+6 \frac{3}{4}=

Amy hikes down a slope to a lake that is

10

2

meters below the trail. Then Amy jumps into the lake, and swims

1

5

meters down. She wonders what her new height is relative to the trail.

\newline

Which of the following equations matches the situation above?

\newline

1

\newline

-10.2+1.5=

\newline

10.2-1.5=

\newline

-10.2-1.5=

Last night, the minimum temperature was

-23^{\circ} \mathrm{C}

5^{\circ} \mathrm{C}

warmer than the lowest temperature where Ailen can safely use her sleeping bag.

\newline

The following equation describes this situation.

\newline

-28+5=-23

\newline

-28

\newline

1

\newline

(A) The temperature dropped

28^{\circ} \mathrm{C}

\newline

(B) The temperature was

28^{\circ} \mathrm{C}

too cold for using the sleeping bag.

\newline

(C) Ailen can safely use her sleeping bag down to

-28^{\circ} \mathrm{C}

Vitor is going to guess on all

40

questions of his multiple-choice test. Each question on the test has

5

answer options.

\newline

Complete the following statement with the best prediction.

\newline

Vitor will correctly answer...

\newline

1

\newline

8

\newline

8

questions but probably not exactly

8

\newline

20

\newline

20

questions but probably not exactly

20

Eiko is wearing a magic ring that increases the power of her healing spell by

30 \%

. Without the ring, her healing spell restores

H

\newline

Which of the following expressions could represent how many health points the spell restores when Eiko is wearing the magic ring?

\newline

2

\newline

0.7 H

\newline

\frac{130}{100} H

\newline

\left(\frac{3}{10}+1\right) H

\newline

H-0.30 H

\newline

30+H

Jose tracks how many times he got fast food each month. He got fast food

12

times in January,

10

times in February,

18

times in March,

4

times in April, and

2

\newline

Find the mean number of times Jose got fast food.

\newline

times getting fast food

Scarlett Squirrel teaches a hula dancing class to young squirrels.

14

squirrels showed up to class on Monday,

10

squirrels on Tuesday,

8

squirrels on Wednesday,

10

squirrels on Thursday, and

12

squirrels on Friday.

\newline

Find the mean number of squirrels. squirrels

Justin did push-ups every weekday this week. He did

8

push-ups on Monday,

14

push-ups on Tuesday,

18

push-ups on Wednesday,

6

push-ups on Thursday, and

8

pushups on Friday.

\newline

Find the mean number of push-ups.

\newline

Seth is planning a route for a train that travels at a constant rate of

220

kilometers per hour. They want to write an equation that shows how many hours

(t)

a trip takes in terms of the trip's distance

(d)

\newline

How should Seth write their equation?

\newline

1

\newline

t=\frac{d}{220}

\newline

\frac{d}{t}=220

\newline

d=220 t

3

\$ 2.91

\newline

Which equation would help determine the cost of

2

\newline

1

\newline

\frac{2}{\$ 2.91}=\frac{x}{3}

\newline

\frac{2}{x}=\frac{3}{\$ 2.91}

\newline

\frac{x}{2}=\frac{3}{\$ 2.91}

\newline

\frac{2}{x}=\frac{\$ 2.91}{3}

\newline

(E) None of the above

5

\$ 6.55

\newline

Which equation would help determine the cost of

4

\newline

1

\newline

\frac{4}{5}=\frac{\$ 6.55}{x}

\newline

\frac{4}{\$ 6.55}=\frac{x}{5}

\newline

\frac{5}{4}=\frac{x}{\$ 6.55}

\newline

\frac{4}{x}=\frac{\$ 6.55}{5}

\newline

(E) None of the above

Sharon tried to solve an equation step by step.

\newline

\begin{array}{l} 9=-3(e-2) \\ 9=-3 e+6 \quad \text { Step } 1 \\ 15=3 e \quad \text { Step 2 } \\ \end{array}

\newline

5=e \quad \text { Step 3 }

\newline

Find Sharon's mistake.

\newline

1

\newline

1

\newline

2

\newline

\operatorname{Step} 3

\newline

(D) Sharon did not make a mistake.

7

watermelons cost

\$ 8.96

\newline

Which equation would help determine the cost of

5

\newline

1

\newline

\frac{x}{5}=\frac{\$ 8.96}{7}

\newline

\frac{x}{5}=\frac{7}{\$ 8.96}

\newline

\frac{7}{5}=\frac{x}{\$ 8.96}

\newline

\frac{5}{x}=\frac{\$ 8.96}{7}

\newline

(E) None of the above

9

\$ 7.74

\newline

Which equation would help determine the cost of

6

\newline

1

\newline

\frac{6}{x}=\frac{9}{\$ 7.74}

\newline

\frac{x}{6}=\frac{9}{\$ 7.74}

\newline

\frac{9}{6}=\frac{x}{\$ 7.74}

\newline

\frac{6}{x}=\frac{\$ 7.74}{9}

\newline

(E) None of the above

3

\$ 4.41

\newline

Which equation would help determine the cost of

4

\newline

1

\newline

\frac{4}{x}=\frac{\$ 4.41}{3}

\newline

\frac{4}{\$ 4.41}=\frac{x}{3}

\newline

\frac{4}{3}=\frac{\$ 4.41}{x}

\newline

\frac{4}{x}=\frac{3}{\$ 4.41}

\newline

(E) None of the above

3

\$ 5.79

\newline

Which equation would help determine the cost of

13

\newline

1

\newline

\frac{13}{\$ 5.79}=\frac{x}{3}

\newline

\frac{x}{13}=\frac{3}{\$ 5.79}

\newline

\frac{3}{\$ 5.79}=\frac{13}{x}

\newline

\frac{13}{x}=\frac{\$ 5.79}{3}

\newline

(E) None of the above

Jessie tried to solve an equation step by step.

\newline

1.5 b+9=11

\newline

1.5 b=3 \quad \text { Step } 1

\newline

b=2 \quad \text { Step } 2

\newline

Find Jessie's mistake.

\newline

1

\newline

1

\newline

2

\newline

(C) Jessie did not make a mistake.

Leon tried to solve an equation step by step.

\newline

\begin{aligned} \frac{1}{2} w-12 & =18 \\ \frac{1}{2} w & =30 \quad \text { Step } 1 \\ w & =15 \quad \text { Step 2 } \end{aligned}

\newline

Find Leon's mistake.

\newline

1

\newline

\operatorname{Step} 1

\newline

2

\newline

(C) Leon did not make a mistake.

Olga tried to solve an equation step by step.

\newline

\begin{array}{rlrl} \frac{1}{4}\left(\frac{1}{3} k+9\right) & =6 & \\ \frac{1}{3} k+9 & =24 \quad \text { Step 1 } \\ \frac{1}{3} k & =15 \quad \text { Step 2 } \\ k & =5 \quad \text { Step 3 } \end{array}

\newline

Find Olga's mistake.

\newline

1

\newline

1

\newline

2

\newline

\operatorname{Step} 3

\newline

D Olga did not make a mistake.

Jeremy and Robin like to collect nickels. Jeremy has

n

nickels, and Robin has

55

nickels. Together they have a total of

100

\newline

Write an equation to describe this situation.

Melissa and Rachel like to make funny hats. Melissa has made

20

zebra-printed hats, and Rachel has made

h

striped hats. Together they have made a total of

42

\newline

Write an equation to describe this situation.

Marcos is bringing

36

cookies to his book club, but he's not sure how many people will be there. He wants to write an equation that shows how many cookies each person will get

(c)

in terms of how many people are there

(n)

\newline

How should Marcos write his equation?

\newline

1

\newline

c=\frac{36}{n}

\newline

n=\frac{36}{c}

\newline

n c=36

Caleb is solving the following equation for

x

\newline

x=\sqrt{x+2}+7

\newline

His first few steps are given below.

\newline

\begin{aligned} x-7 & =\sqrt{x+2} \\ (x-7)^{2} & =(\sqrt{x+2})^{2} \\ x^{2}-14 x+49 & =x+2 \end{aligned}

\newline

Is it necessary for Caleb to check his answers for extraneous solutions?

\newline

1

\newline

\newline

...

...