Modeling with Systems of Linear Equations

Bethany has $3.10 in nickels and dimes. She has a total of 44 coins. What is the value of the nickels Bethany has?

$1.30

An aircraft travels with the wind for 120 miles in 0.75 of an hour. The return trip is flown against the wind and takes exactly 1 hour. Which system of linear equations represents x, the speed of the plane in miles per hour, and y, the speed of the wind in miles per hour? Recall the formula d = rt.

0.75(x + y) = 120 x − y = 120

Micah rows his boat on a river 4.48 miles downstream, with the current, in 0.32 hours. He rows back upstream the same distance, against the current, in 0.56 hours. Assuming his rowing speed and the speed of the current are constant, what is the speed of the current?

3 miles per hour

Zorah, a musician, pays $120 to have her instrument tuned and $10 per hour for a booth at a fair. She estimates that she earns $25 per hour in tips. The equation can be used to represent the break-even point. 120 + 10x = 25x How many hours, x, will Zorah have to play in order to break even?

8

A store sells both cold and hot beverages. Cold beverages, c, cost $1.50, while hot beverages, h, cost $2.00. On Saturday, drink receipts totaled $360, and 4 times as many cold beverages were sold than hot beverages. Which system of linear equations represents the beverage sales on Saturday?

c = 4h 1.5c + 2h = 360

The table below models the cost, y, of using a high-efficiency washing machine and a standard washing machine over x number of years. Which equation represents the cost of the high-efficiency washing machine over a given number of years? Which equation represents the cost of the standard washing machine over a given number of years? After how many years of use would the washing machines cost the same amount? Which washing machine would be the more practical purchase if kept for 9 years?

1. y=25x+500 2. y = 30x + 400X 3. 20 4. standard machine

Mr. Yi buys vegetables at a market. He purchases 6 pounds of potatoes, p, and 3 pounds of onions, n, for $18. Onions cost twice as much as potatoes. To determine the unit price for each item, his daughter sets up and solves the system of equations shown. 6p + 3n = 18 and 2n = p 6(2n) + 3n = 18 12n + 3n = 18 15n = 18; n = $1.20 Onions cost $1.20 per pound. Analyze the daughter's solution. Which statements are true? Check all that apply.

The equation 2n = p should be 2p = n. The actual cost of the onions is $3.00 per pound. Potatoes cost $1.50 per pound.

Charlotte's weekly paycheck is based on the number of hours worked during the week and on the weekend. If she works 13 hours during the week and 14 hours on the weekend, she earns $250.90. If she works 15 hours during the week and 8 hours on the weekend, she earns $204.70. How much more does Charlotte earn per hour on the weekends than she earns during the week? Round to the nearest cent.

$2.30

Bryan's monthly electric bill is determined by adding a flat administration fee to the product of the number of kilowatt hours of electricity used and the cost per kilowatt hour. When he uses 1,100 kilowatt hours of electricity, his bill is $113. When he uses 1,500 kilowatt hours of electricity, his bill is $153. What is the monthly administration fee?

$3

A factory has fixed costs of $1,275 per month. The cost of producing each unit of its product is $2.50. Each unit sells for $10. The equations that model the cost and revenue for producing x units are shown. Cost: C(x) = 1275 + 2.5x Revenue: R(x) = 10x If the company sells every unit it produces, how many units must it sell in order to break even?

170

A business owner pays $1,200 per month in rent and a total of $120 per hour in employee salary for each hour the store is open. On average, the store brings in $200 in net sales per hour. Which equation can be solved to determine the break-even point if x represents the number of hours per month the store is open?

200x - 120x = 1,200

Dalia flies an ultralight plane with a tailwind to a nearby town in 1/3 of an hour. On the return trip, she travels the same distance in 3/5 of an hour. What is the average rate of speed of the wind and the average rate of speed of the plane? Initial trip: Return trip: Let x be the average airspeed of the plane. Let y be the average wind speed. Initial trip: 18 = (x + y) Return trip: 18 = (x - y) Dalia had an average airspeed of miles per hour. The average wind speed was miles per hour.

42, 12

A graphics company charges $50 per hour to create a logo plus $250 for the ownership rights of the logo. A private contractor charges $75 per hour to develop a logo plus a $100 supply fee. Which equation can be solved to determine x, the number of hours after which the costs would be the same?

50x + 250 = 75x + 100

Enrique has $50 in his lunch account and spends $5 per day from the account. Maya has $46 in her lunch account and spends $4 per day from the account. Which equations model the situation?

A. 50 - 5x = y and 46 - 4x = y

Danae is choosing between two jobs. One job pays an annual bonus of $1,500 plus $120 per day worked. The second job pays an annual bonus of $2,500 plus $110 per day worked. Which equation can be solved to determine after how many days, d, Danae would make the same amount of money regardless of the job she chooses?

C. 120d + 1,500 = 110d + 2,500

Lily sold 18 items at the street fair. She sold bracelets for $6 each and necklaces for $5 each for a total of $101. Which system of equations can be used to find b, the number of bracelets she sold, and n, the number of necklaces she sold?

b + n = 18 6b + 5n = 101

Nina is 10 years younger than Deepak. Deepak is 3 times as old as Nina. Which system of equations can be used to find d, Deepak's age, and n, Nina's age?

d = n + 10 d = 3n

At his new job, Jeremiah can choose an hourly rate of $9 plus a $50 weekly bonus for opening the store, or an hourly rate of $10 per hour with no opening bonus. The equations model his salary options. y = 9x + 50 y = 10x What does x represent?

the number of hours worked per week