Section 11.1Systems of Linear Equations; Substitution and Elimination

A movie theater sells tickets for $8.00 each, with seniors receiving a discount of $2.00. One evening the theater took in $3580 in revenue. If x represents the number of tickets sold at $8.00 and y the number of tickets sold at the discounted price of $6.00, write an equation that relates these variables.

Suppose we also know that 525 tickets were sold. Write another equation relating the variables x and y.

2 3 3Graph the system:

4 3x yx y

2 13

y x

4 3y x

x

y

3 9The 2 lines intersect at the point ,7 7

which is the solution to the system of equations.

3 9,7 7

2 3 3Solve:

4 3x yx y

4 3y x

2 333 4xx

4 3y x

2 12 9 3x x 6 314 6

14 7x x

947

37

3y

2 3 3 43 9 3 97 7 7 7

3

Solution:3 9,7 7

2 3 1Solve:

2 3x yx y

3

3 12 33

2x yx y

2 3 16 3 9x yx y

8 8x

1x

2 1 3y

1y

2 3 1 21 1 1 1 3

Solution: 1, 1

3 2Solve:

2 6 1x yx y

3 2x y 3 2x y

3 22 6 1yy

6 4 6 1y y

4 1

Since this statement is false we conclude there is no solution. We say the system is inconsistent.

x

y

3 2Solve:

2 6 4x yx y

3 2x y

22 3 6 4yy

6 4 6 4y y

4 4

Since this statement is true but we have no variables, the two equations are equivalent so the equations are dependent.

3 2x y

x

y

Solution: , 3 2x y x y

Use the method of elimination to solve the system of equations.2 4

3 2 2 102 3 7

x y zx y zx y z

2 4

2 2 3 72x y zx y z

2 4 2 4 6 14

x y zx y z

5 5 10y z

3 2 2 10 3 2 3 7 3

x y zx y z

3 2 2 10 3 6 9 21

x y zx y z

4 7 11y z 45 5 104 y z

4 7 5 115 y z

20 20 40 y z

20 35 55 y z

15 15 z 1 z 5 5 1 10y 1y

2 1 3 1 7x 2x Solution: 2, 1,1

2 5 3

34 1

3 x y zx y z

2 3 0Solve: 2 5

3 4 1

x y zx y zx y z

2 3 0

2 2 52x y zx y z

2 3 0 2 4 2 10

x y zx y z

10y z

2 2 16y z

3 6 3 15 3 4 1

x y zx y z

2 2 10y z

0 4

2 2 20 y z

Since this statement is false we conclude there is no solution. We say the system is inconsistent.

2 1Solve: 2 2

4 5 4

x y zx y zx y z

2 1 2

22

2 x y zx y z

2 2 4 2

2 2x y zx y z

3 3 0y z

2 1 4 4

45

4 x y zx y z

3 3 0y z

4 4 8 4 4 5 4

x y zx y z

0 0Since this statement is true but we have no variables, the equations are dependent.

3 3 0 so y z y z

2 1 so 3 1x y z x z

, , 3 1, , is any real numberx y z x z y z z

46

a b ca b c

2 2 2a c

4

9 3 0

3 3a b c

a b c

3 3 3 12 9 3 0a b ca b c

12 4 12a c

4 4 4a c 8 16a

2a

1 3c a

6 5b a c

2a 1 3c a

6 5b a c