AlgebraAlgebra7.4 7.4

Applications ofApplications of Linear SystemsLinear Systems

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

Let’s briefly review each method.Let’s briefly review each method.

The solution is the point of intersection of the two graphed lines.

1. Graph and Check Method

To solve:1. Graph both equations2. Identify intersection point (x,y)3. Plug in to original equations to check

3x + y = 52x – y = 10

y = 5 – 3x

2x – ( ) = 105 – 3x

2x – 5 + 3x = 10

5x – 5 = 10

5x = 15

x = 3

y = 5 – 3(3)

y = 5 - 9

y = - 4

The solution is (3, - 4)

2. Substitution Method

3x + 3y = 9

5x + 2y = 12[ ]-3

[ ] 2 6x + 6y = 18

Linear Combinations Method

-15x - 6y = -36

-9x = -18x = 23x + 3y = 9

3(2) + 3y = 96 + 3y = 9

3y = 3y = 1

Solution: (2, 1)

2

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 = 246x – y = 18

Linear combinations

x = 2y + 42x - 6y = 12

Substitution

x + y = 126x + y = 32

Either

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 mphLet y be the # of hours you sprinted at 6 mph

Quantity:x + y = 3Value: 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 soldLet y be the # of Asics sold

Quantity:x + y = 28Value: 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