Honors Algebra 2: Systems of 2 Linear Equations Word Problems

Calculator Active. Answer to the nearest hundredth where appropriate.

  1. An airplane flying with a headwind traveled 1000 miles from one city to another in 2 hours and 12 minutes. On the return flight, flying with a tail wind, the total time was only 2 hours. Find the average air speed of the plane and the speed of the wind.
    2. Solution The plane flew at an average speed of 477.27 mph, and the wind's speed was 22.73 mph.
  2. A swimmer can swim 18 miles downstream in a nearby river in 3 hours. However, the return trip upstream takes him 6 hours. Find the swimmer's average speed in still water and the speed of the river's current.
    3. Solution The swimmer's average speed in still water was 4.5 mph, and the current's speed was 1.5 mph.
  3. Two airplanes fly in opposite directions from the same airport. The same plane leaves a half-hour after the first. The second plane travels at a rate of 60 mph faster than the first. Find the air speed of each plane if two hours after the first plane starts, the two planes are 2015 miles apart.
    4. Solution The way I do it, this problem ends up being in one variable.
    Rate in miles/hr * time in hrs = dist in mi
    First
    \(r\)
    2
    \(2r\)
    Second
    \(60+r\)
    1.5
    \(1.5(60 + r)\)

    The planes are 2015 miles apart, so \(2r + 1.5(60 + r) = 2015\). Then \(2r + 90 + 1.5r = 2015\) so \(3.5r = 1925\), and \(r = 550\) mph. The air speed of the second plane would be 610 mph.
  4. The admission fee at a small fair is $4.00 for children and $1.50 for adults (which, of course, is unfair.) On a certain day, 2200 people enter the fair, and $5050 is collected. How many children attended?
    5. Solution There were 700 children.
  5. The school that Stefan goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. Find the price of a senior citizen ticket and the price of a child ticket.
    6. Solution Seniors are $8 per ticket. Children are $14 per ticket.
  6. The sum of the digits of a certain two-digit number is 7. Reversing its digits increases the number by 9. What is the number?
    7. Solution 34
  7. The state fair 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 8 vans and 8 buses with 240 students. High School B rented and filled 4 vans and 1 bus with 54 students. Every van had the same number of students in it as did the buses. Find the number of students in each van and in each bus.
    8. Solution There were 8 vans and 22 busses.
  8. The senior classes at High School A and High School B planned separate trips to New York City. The senior class at High School A rented and filled 1 van and 6 buses with 372 students. High School B rented and filled 4 vans and 12 buses with 780 students. Each van and each bus carried the same number of students. How many students can a van carry? How many students can a bus carry?
    9. Solution There were 18 vans and 59 busses.
  9. A boat traveled 336 miles downstream and back. The trip downstream took 12 hours. The trip back took 14 hours. What is the speed of the boat in still water? What is the speed of the current?
    10. Solution The boat went 26 mph, the current was 2 mph.
  10. DeShawn and Shayna are selling flower bulbs for a school fundraiser. Customers can buy bags of windflower bulbs and bags of daffodil bulbs. DeShawn sold 10 bags of windflower bulbs and 12 bags of daffodil bulbs for a total of $380. Shayna sold 6 bags of windflower bulbs and 8 bags of daffodil bulbs for a total of $244. What is the cost each of one bag of windflower bulbs and one bag of daffodil bulbs?
    11. Solution bag of windflower bulbs was $14, bag of daffodil bulb was $20.
  11. How many gallons of a 20% alcohol solution and a 50% alcohol solution must be mixed to get 9 gallons of a 30% alcohol solution?
    12. Solution 6 gal. of 20% and 3 gal of 50%.