Applications for Systems of Equations

Algebra I

Example #1

Flying to Ankara with a tailwind a plane averaged 368 mph. On the return trip the plane only averaged 332 mph while flying back into the same wind. Find the speed of the plane in still air and the speed of the wind.

Solution: let x be the planes speed and y the winds speedx + y = 368 (With the wind)x – y = 332 (Against the wind)

Answer: Plane 350mph Wind 18mph

Example #2

Traveling with the current a certain boat went 19 mph. Against the same current the boat only went 3 mph. What’ s the speed of the current? What’s the speed of the boat in still water?

Solution: Let x = boats speed and y = currents speedx + y = 19(with current)x – y = 3 (against current)

Answer: Boat 11mph, Current 8mph

Example #3 The difference of two numbers is 1 Their sum

is 15. Find the numbers. Solution: Let x and y be the numbers

x – y = 1x + y = 15

Answer: 7 and 8

Example #4 The sum of the digits of a certain two-digit

number is 10. When you reverse the digits you increase the number by 54. What’s the number?

Solution: Let x be the tens digit and y be the ones digit.x + y = 10(10y + x) – (10x + y) = 54

OR Use guess and check to find the solution.

Answer: 28

Example #5 A boat traveled 312 miles downstream and

back. The trip down stream took 12 hours. The trip back took 39 hours. What is the speed of the boat in still water? What is the speed of the current?

Solution: Let x = boats speed and y = currents speedremember d=rt12(x + y) = 31239(x – y) = 312

Answer: Boat 17mph, Current 9mph

Example #6 New York City is a popular field trip destination. This

year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 9 vans and 14 buses with 739 students. High School B rented and filled 5 vans and 12 buses with 609 students. Every van had the same number of students as did the buses. Find the number of students in each van and in each bus.

Solution: Let x = students in vans and y = students in buses9x + 14y = 7395x + 12y = 609

Answer: 9 students per van and 47 per bus.

Example #7 Krystal’s school is selling tickets to the annual

talent show. On the first day of ticket sales the school sold 6 senior citizens tickets and 6 child tickets for a total of $138. the school took in $151 on the second day by selling 7 senior citizens tickets and 6 child tickets. Find the price of a senior ticket and the price of a child ticket.

Solution: Let x = price for a seniors ticket and y = price for a child’s ticket6x + 6y = 1387x + 6y = 151

Answer: $13 for a senior ticket and $10 for a child’s ticket