Applications of Linear Systems of Equations

1. Applications of Linear Systems of Equations Many problems involve more than one unknown quantity. Although some problems with two unknowns can be solved using just one variable, many times it is easier to use two variables. To solve a problem with two unknowns, we write two equations that relate the unknown quantities. The system formed by the pair of equations then can be solved using the previous methods. Procedure: Solving an Applied Problem by Writing a System of Equations Step 1. Determine what you are to find. Assign a variable for each unknown and write down what it represents. Step 2. Write down other information. If appropriate, draw a figure or a diagram and label it using the variables from Step 1. Use a chart or a box to summarize the information. Step 3. Write a system of equations. Write as many equations as there are unknowns. Step 4. Solve the system. Step 5. Answer the question(s). Be sure you have answered all questions posed. Step 6. Check. Check your solutions in the original problem. Be sure your answer makes sense.

2. Example 1. Solve by using a system of equations: The sum of two numbers is 37 and their difference is –3. Solution: Answer: The two numbers are 17 and 20. Their sum is 37 and their difference is –3. Your Turn Problem #11 Solve by using a system of equations: The sum of two numbers is 51 and their difference is 15. Let x = first number and let y = second number We need two equations with two unknowns. Then use either method to solve the system Answer: The two numbers are 33 and 18.

3. Example 2. Solve by using a system of equations: Laura and Tony are planning to move and need to buy some cardboard boxes. At Smart and Final, they can buy either 10 small and 12 medium boxes fro $42, or 5 small and 10 medium boxes for $29. Find the cost of each size box. Solution: Let x = cost of a small box and let y = cost of a medium box We need two equations with two unknowns. Answer: The cost of a small box is $1.80 and the cost of a medium box is $2.00. Your Turn Problem #2 Solve by using a system of equations: At the Rancho Nut Ranch, 5 pounds of peanuts and 6 pounds of cashews cost $70, and 3 pounds of peanuts and 7 pounds of cashews cost $76. Find the prices charged by the ranch for a pound of peanuts and a pound of cashews. Then use either method to solve the system Answer: Peanuts are $2.00 per pound Cashews are $10.00 per pound

4. Example 3. Solve by using a system of equations: Tickets to a play at Cucamonga Jr. High cost $2.50 for general admission and $2.00 with a student identification. If 92 people paid to see a performance and $203 was collected, how many of each type of admission were sold? Let x = number of general admission tickets sold let y = number of student tickets sold Solution: We need two equations with two unknowns. Let’s use substitution. Answer: # of generals admission = 38 # of students = 54 Your Turn Problem #3 Solve by using a system of equations: On a 10-day trip Dan rented a car for $31 a day at weekend rates and $40 a day at weekday rates. If his rental bill was $364, how many days did he rent a each rate? The End. B.R. 1-15-07 Answer: Dan rented the car for 4 weekend days and 6 weekdays.