Ants, Freeways, & Other Systems

Solving linear Systems AlgebraicallyState Standard – 2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, linear combination, with graphs, or with matrices.

Created byJason L. Bradbury

The Linear Combination Method

1) Multiply one or both of the equations by a constant.

2) Add the revised equations in order to eliminate one of the variables.

3) Substitute the value in Step 2 into one of the original equations to get the other variable

( , )

Example 1

Solve using the Linear Combination method:

2x – 4y = 13

4x – 5y = 8

-2( )2x – 4y = 134x – 5y = 8

-4x + 8y = -264x – 5y = 8

3y = -18

y = -6

2x – 4(-6) = 13

2x + 24 = 13- 24 -24

2x = -11

x = -112

-112

-6

( , )

Example 2

Solve using the Linear Combination method:

2x + 3y = -1

-5x + 5y = 15

5( )2x + 3y = -1-5x + 5y = 15

10x + 15y = -5-10x + 10y = 30

25y = 25

y = 1

2x + 3(1) = -1

2x + 3 = -1- 3 -3

2x = -4

x = -2

-2 1

2( )

Example 3

A caterer is planning a party for 64 people. The customer has $150 to spend. A $39 pan of pasta feeds 14 people and a $12 sandwich tray feeds 6 people. How many pans of pasta and how many sandwich trays should the caterer make?

-2( )14p + 6s = 6439p + 12s = 150

-28p – 12s = -12839p + 12s = 150

11p = 22

p = 2

14(2)+ 6s = 64

28 + 6s = 64- 28 -28

6s = 36s = 6

The caterer should make 2 pans of pasta and 6 trays of sandwiches.

Guided Practice6x + 6y = 3

4x + 4y = 2

3x – 3y = 3

-4x + y = 21

-2x + y = 13

x – 4y = -31

x – 6y = 6

-3x + 2y = -2

-5x + 7y = 10

15x – 21y = 22

-4x + 8y = 24

-x + 2y = 6

If you get the variables to cancel and you get: 0 = 0

You will have: Infinitely many solutions

No solutions

If you get the variables to cancel and you get: 0 = (some #)

You will have:

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