Word problems that lead to Section 2 Problem 2. A total of 925 tickets were sold for $5,925. If adult tickets cost $7.50, and children's tickets cost $3.00, how many tickets of each kind were sold? (Compare Example 1.) To see the answer, pass your mouse over the colored area. Let x be the number of adult tickets. Let y be the number of childeren's tickets. Here are the equations:
In equation 2), make the coefficients into whole numbers by multiplying both sides of the equation by 10:
To eliminate y, for example: Multiply equation 1) by −30, and add. The solution is: x = 700, y = 225. Problem 3. Mr. B. has $20,000 to invest. He invests part at 6%, the rest at 7%, and he earns $1,280 interest. How much did he invest at each rate? (Compare Example 2.) Let x be how much he inveted at 6%. Let y be how much he inveted at 7%. Here are the equations:
To eliminate x, for example, from equations 1) and 2'): ,Multiply equation 1) by −6, and add. The solution is: x = $12,000. y = $8,000. Problem 4. Edgar has 20 dimes and nickels, which together total $1.40. How many of each does he have? (Compare Problem 1.) Let x be the number of dimes. Let y be be the number of nickels. Here are the equations:
To eliminate x, for example, from equations 1) and 2'), multiply equation 1) by −10, and add. The solution is: x = 8 dimes. y = 12 nickels. Problem 5. How many gallons of 20% alcohol solution and how many of 50% alcohol solution must be mixed to produce 9 gallons of 30% alcohol solution? (Compare Example 3.) (9 gallons of 30% alcohol solution = .3 × 9 = 2.7 gallons of pure alcohol.) Let x be the number of gallons of 20% solution. Let y be the number of gallons of 50% solution. Here are the equations:
To eliminate x, for example, from equations 1) and 2'), multiply equation 1) by −2, and add. The solution is: x = 6 gallons. y = 3 gallons. Problem 6. 15 gallons of 16% disenfectant solution is to be made from 20% and 14% solutions. How much of those solutions should be used? (15 gallons of 16% solution = .16 × 15 = 2.4 gallons of pure disenfectant.) Let x be the number of gallons of 20% solution. Let y be the number of gallons of 14% solution. Here are the equations:
To eliminate x, for example, from equations 1) and 2'), multiply equation 1) by −20, and add. The solution is: x = 5 gallons. y = 10 gallons. Problem 7. It takes a boat 2 hours to travel 24 miles downstream and 3 hours to travel 18 miles upstream. What is the speed of the boat in still water, and how fast is the current? (Compare Example 4.) Let x be the speed of the boat in still water. Let y be the speed of the current. Here are the equations:
To eliminate y, simply add the equations. The solution is: x = 9 mph. y = 3 mph. Problem 8. An airplane covers a distance of 1500 miles in 3 hours
wind. What is the speed of the plane in still air? (Compare Example 4.) Let x be the speed of the plane in still air. Let y be the speed of the wind. Here are the equations:
To eliminate y, simply add the equations. The solution is: x = 475 mph. Next Lesson: Quadratic equations Please make a donation to keep TheMathPage online. Copyright © 2001-2007 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |