7.4 Applications of Linear Systems Algebra 7.4 Applications of Linear Systems

There are 3 methods for solving systems of equations There are 3 methods for solving systems of equations. The method you choose should be which one is the easiest in the given case. Let’s briefly review each method.

The solution is the point of intersection of the two graphed lines. 1. Graph and Check Method The solution is the point of intersection of the two graphed lines. To solve: Graph both equations Identify intersection point (x,y) Plug in to original equations to check

2. Substitution Method 3x + y = 5 y = 5 – 3x 2x – y = 10 2x – ( ) = 10 The solution is (3, - 4) x = 3

Linear Combinations Method 6x + 6y = 18 [ ] 2 3x + 3y = 9 5x + 2y = 12 -15x - 6y = -36 [ ]-3 -9x = -18 x = 2 2 3x + 3y = 9 3(2) + 3y = 9 Solution: (2, 1) 6 + 3y = 9 3y = 3 y = 1

When to Use Which Method Graphing: When the equation is already graphed and the intersection (x,y) is whole numbers Substitution: When one variable is already isolated Linear combinations: No variable has coefficient of 1 or -1 and columns are lined up

Which method would you use? 8x + y = 24 6x – y = 18 Linear combinations x = 2y + 4 2x - 6y = 12 Substitution Either x + y = 12 6x + y = 32

Set-up for Mixture Problems Write one equation to describe QUANTITY. Write other equation to describe VALUE.

Set up a system and solve the mixture problem. You exercised on a treadmill for 3 hours. You ran at 4 miles per hour, then you sprinted at 6 miles per hour. If the treadmill monitor says that you ran and sprinted 14 miles, how long did you run at each speed? Let x be the # of hours you ran at 4 mph Let y be the # of hours you sprinted at 6 mph Quantity: x + y = 3 Value: 4x + 6y = 14 You ran (x) for 2 hours and you sprinted (y) for 1 hour. Now, you solve.

Set up a system and solve the mixture problem. A store sold 28 pairs of running shoes for a total cost of $2220. Nikes sold for $70 per pair and Asics sold for $90 per pair. How many of each style were sold? Let x be the # of Nikes sold Let y be the # of Asics sold Quantity: x + y = 28 Value: 70x + 90y = 2220 The store sold 15 pairs of Nikes and 13 pairs of Asics. Now, you solve.

Together from the Homework pg. 422 #43, 47

Homework pg. 421 #25 – 45 odd, 46 - 50