Systems of Equations – DOK 3
Directions: Write a system of equations to describe the situation below and solve using substitution.
1) Mika and her friend Hannah are knitting scarves for the homeless. Mika had already completed 3 scarves and has committed to making 3 more scarves per day. Hannah, who hasn't completed any scarves yet, has more free time and can make 4 scarves per day. At some point, Hannah will catch up and they will both have completed the same number of scarves. How many scarves will each woman have finished? How long will that take?
2) Mr. Li, the owner of two car dealerships in Newberg, is holding a contest to see which one can sell the most cars. Newberg Cars has already sold 12 cars, and Li's Autos has sold 14 cars. Going forward, the salespeople at Newberg Cars think they can sell 8 cars per day, whereas the salespeople at Li's Autos are aiming for sales of 6 cars per day. If the salespeople's predictions are accurate, it won't be long before the two dealerships are tied. How many cars will each lot have sold? How long will that take?
3) A charity is considering the possibility of having a benefit night at two different restaurants. The owner of the local Italian restaurant has offered to make a donation of$307and$6per diner that night. On the other hand, the owner of the Mexican restaurant has said he could contribute$603plus$2per diner. Based on the number of diners who have promised to participate in the event, it appears that each restaurant would donate the same total amount. How many diners promised to participate? How much would each restaurant donate?
4) Ethan and Brooke both went bowling last weekend, but not together. Ethan went during open bowling time, paying$12for shoe rental and$3per hour. Brooke went on family night, when each hour was$4and shoe rental was$9. As it turned out, they played for the same number of hours and spent the same amount. How much did each one spend? How many hours did each one bowl?
5) Zachary is a kickboxing instructor who will be teaching classes at a local gym. To get certified as an instructor, he spent a total of$96. Zachary will be earning a base salary of$90per month from the gym, plus an additional$3for every class he teaches. If Zachary teaches a certain number of classes during his first month as an instructor, he will earn back the amount he spent on certification. How many classes will that take? How much will Zachary's expenses and earnings be?
6) Carl is starting a business selling handmade necklaces. He has decided to invest an initial amount of$468for advertising, and materials cost$4for each necklace he makes. Carl can sell his creations for$40per necklace. Once he makes and sells a certain number of necklaces, he will break even, with identical expenses and sales. How many necklaces would that take? What would the total expenses and sales be then?
7) A rental car company is running two specials. Customers can pay$43to rent a compact car for the first day plus$4for each additional day, or they can rent the same car for$51the first day and$3for every additional day beyond that. Oscar notices that, given the number of additional days he wants to rent the car for, the two specials are equivalent. How much would Oscar pay in total? How many additional days does Oscar want?
8) The volleyball team and the wrestling team at Newton High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets$5per car. In addition, they have already brought in$17from past fundraisers. The wrestling team has raised$99in the past, and they are making$3per 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?
9) Mr. Hunt is contemplating which chauffeured car service to take to the airport. The first costs$30up front and$1per mile. The second costs$2plus$3per mile. For a certain driving distance, the two companies charge the same total fare. What is the total fare? What is the distance?
10) Cindy is training for a triathlon and needs to swim a certain distance for today's workout. If she swims at the rec center pool, she will complete a 259-yard warmup and then swim laps in a lane that is 42 yards long. If Cindy swims at the indoor pool at her gym, she will complete a 248-yard warmup, plus a main set that consists of 43 yards per lap. If Cindy swims the correct number of laps, she can complete the same distance in either pool. How many laps will that take? How long will Cindy's workout be in total?
11) Two friends, Kristen and Belle, are growing out their hair. They plan to cut it off at a certain point and donate it to a charity that makes wigs for people with cancer. Kristen's hair is already 16 centimeters long and grows at a constant rate of 3 centimeters per month. Belle's hair is 36 centimeters and growing at a speed of 2 centimeters per month. If the girls get their hair cut on a certain day, they will have exactly the same length to donate. How long will their hair be? How long will that take?
12) A fashion photographer needs to hire a stylist to prepare her models. Ronald charges$180for showing up, plus$74per hour. Sasha charges$189plus$71per hour. The photographer realizes that, given the expected duration of her photo shoot, either stylist would cost her the same amount. What would the duration be? What would the cost be?
13) Harold and his mom are planting vegetables in their garden. Harold has finished planting 1 row of carrots so far and is planting new rows at a rate of 5 rows per hour. His mom has finished 11 rows of tomatoes and will continue planting at 3 rows per hour. Once the pair get to the point where they have finished the same number of rows, they will take a break and decide what to plant next. How many rows will they each have completed? How long will that take?
14) Layla has a home-based business making and selling scented soaps. She intially spent$56to purchase soap-making equipment, and the materials for each pound of soap cost$8. Layla sells the soap for$10per pound. Eventually, she will sell enough soap to cover the cost of the equipment. What will be Layla's total sales and costs be? How much soap will that be?
15) Two classmates have decided to read all the volumes in a popular series of books. Olivia has already read 8 volumes and will continue to read new ones at a rate of 2 volumes per week. Lisa, who hasn't started reading the series yet, will read 3 volumes per week. At some point, Lisa will catch up with Olivia and they will be reading the same book. How long will that take? How many volumes will each girl have read by then?
16) Maura and Zack, the boy she was babysitting, were playing basketball together. Her score was 18 points, and his score was 20 points. Maura wanted to make the game more fair, so she called a time-out and modified the rules a bit. Maura explained that, for the rest of the game, she would get 3 points per basket, and Zack would get 1 point per basket. Then they played a bit longer. After the time-out, they both made the same number of baskets and ended up with a tied score. How many baskets did each person make after the time out? How many points did each person have at the end?
17) Mrs. Warren wants to get family portraits taken and is comparing prices between two different photography studios. Cooper Photography charges$19per portrait sheet, plus$41for the session fee. Lasting Memories Company charges$45for the session fee and$15per portrait sheet. If Mrs. Warren plans to purchase a certain number of portrait sheets, the cost will be the same at either studio. How many portrait sheets would that be? What would the total cost be?
18) Layla and Owen are making presentations for a class project. Layla's slideshow starts with a verbal introduction that is 4 seconds long, and then each slide is left up for 5 seconds. Owen leaves each slide onscreen for 4 seconds, and his introduction lasts 18 seconds. Layla and Owen notice that their presentations have both the same number of slides and the same duration. How many slides are in each presentation? How long is each presentation?
19) A couple of friends decide to race each other. Hayley can run 7 yards per second, whereas Sasha can run 8 yards per second. Because she is slower, Hayley also gets a head start of 19 yards. Shortly after they start running, Sasha will catch up to Hayley. How far will Sasha have to run? How long will that take?
20) Diana is comparing the cost of a fresh lobster dinner at two different restaurants. The first restaurant charges$45for the meal, plus$1per pound for the lobster she picks. At the second restaurant, Diana would pay$3per pound for the lobster, in addition to$39for the meal. Diana realizes that, in theory, dinner at both restaurants could cost the same amount if the lobster had a certain weight. What is the weight? How much would Diana pay for her dinner?
Answers:
1) Each woman will have finished 12 scarves in 3 days
2) The dealerships will each have sold 20 cars in 1 day.
3) With 74 diners promising to participate, each restaurant would donate$751.
4) Ethan and Brooke each spent$21and bowled for 3 hours
5) Once Zachary teaches 2 classes, his expenses and earnings will both be$96.
6) After selling 13 necklaces, Carl would have total expenses and sales of$520.
7) Oscar would pay$75either way for 8 additional days.
8) Each team will have raised a total of$222after washing 41 cars.
9) The total fare is$44for a driving distance of 14 miles.
10) If Cindy swims 11 laps at either pool, she can complete a total workout of 721 yards.
11) Their hair will be 76 centimeters in 20 months
12) If the shoot lasted for 3 hours, either stylist would cost$402.
13) Harold and his mom will each be done planting 26 rows in 5 hours.
14) Layla's sales and costs will both be$280once she sells 28 pounds of soap.
15) In 8 weeks, the girls will each have read 24 volumes in the series.
16) Maura and Zack each made 1 basket after the time-out, for a score of 21
17) If Mrs. Warren orders 1 portrait sheet, the portrait session will cost$60at either studio
18) Each presentation has 14 slides and is 74 seconds long.
19) Sasha will catch up to Hayley after running 152 yards in 19 seconds
20) If the lobster weighed 3 pounds, dinner at either restaurant would cost$48.