+ Water wars An enemy submarine has launched a missile toward another submarine in your fleet...

+Water wars

An enemy submarine has launched a missile toward another submarine in your fleet following the path 2x-y=4. Your submarine retaliates launching a counter missile following the path x-2y=-1. How would you determine if your missile was successful in its mission?

+

Solve Linear Systems by Graphing, Substitution, and Elimination3.1

+Graph to find the solution

Y=-1

3x+y=5

+Solve the system of equations - graphing

(-5,0)

+Solve the system of equations - graphing

(-8,5)

3 4 4

2 2

x y

x y

+Questions to ponder …

Will two lines always intersect?

What would you determine about the solution to the system?

Are there any other scenarios when two lines are graphed?

What would you determine about the solution to the system?

+Graph to find the solution

y=3x+2

-3x+y=-2

+Graph to find the solution

y=4x+3

20x-5y=-15

+Assignment

Solve by graphing with the graphing calculator.

P. 142 (30-35, 59-61)

+ Systems of equations

Solve by substitution

3 4 4

2 2

x y

x y

1. Pick one equation and isolate a variable.

2 2x y

2. Replace that variable with its value in the other equation

3 4 4x y 3( 2 2) 4 4y y

3. Solve for the remaining variable .

6 6 4 4y y 2 10y

5y

Best to choose a variable that has a coefficient of one.

4. Find the value of the other variable by plugging the answer back into one of the equations.

2 2x y

2 5 2x

10 2x

8x

8,55. Write solution as an ordered pair.

Remember this is the point where the lines intersect.a.k.a--solution to the system (satisfies both equations).

+Solve the following system of equations by substitution.

3 13

2 2 10

x y

x y

3 13y x

3 13y x

2 2 3 13 10x x 2 6 26 10x x 8 26 10x

8 16x 2x

3 2 13y

7y

2, 7

+Water wars

An enemy submarine has launched a missile toward another submarine in your fleet following the path 2x-y=4. Your submarine retaliates launching a counter missile following the path x-2y=-1. How would you determine if your missile was successful in its mission?

+Solve the system of equations - substitution

(2,1)

+Solving by Substitution

x-7y=6

-3x+21y=18

+Solving by Substitution

Y=2x-1

-6x+3y=-3

+Systems of Equations

Solve by Linear Combination (Elimination)

2 4 13

4 5 8

x y

x y

1. Pick one variable -doesn’t matter which variable

umm - I pick x

2. Multiply one or both equations so that the variable you chose are opposites

Multiply eq. 1 by -2

4 8 26x y to obtain

Leave eq. 2 the same 4 5 8x y 3. Add the two equations.

3 18y 6y

4. Solve for the remaining variable.

5. Evaluate the other variable by substitution. 4 5 8x y

4 5 6 8x 4 30 8x 4 22x

11

2x

6. Write the solution as an ordered pair.11

, 62

Solve the following system of equations by elimination.

2 3 1

5 5 15

x y

x y

x 5

x 210 15 5x y 10 10 30x y

25 25y 1y 5 5 1 15x

5 10x 2x 2,1

What would I have needed to do if I wanted to eliminate the y first?

--multiply eq. 1 by 5

--multiply eq. 2 by -3

Solve the following system of equations by substitution or elimination.

2 3

2 4 7

x y

x y

2 3x y I chose substitution!!!

2 2 3 4 7y y 4 6 4 7y y

6 7

What can you tell me about the lines?

--they do not intersect (parallel)

Solution to the system

Solve the following system of equations by substitution or elimination.

6 10 12

15 25 30

x y

x y

I chose elimination!!!

x 5

x 230 50 60x y 30 50 60x y

0 0

What can you tell me about the lines?--same line!!!

Give one possible solution. 2,0

How many solutions are possible?

Is any ordered pair a solution? NO!!

Why is this a possible solution? Satisfies the equation!

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 $12sandwich tray feeds 6 people. How many pans and how manysandwich trays should the caterer make?

Write a system of equations to represent the information in the problem.

How many unknowns are there?

2x -- number of pans of pasta

y -- number of sandwich trays

14 6 64x y 39 12 150x y

Solve the system by substitution or elimination.

2 pans of pasta

6 sandwich trays

x

y

+

Solve by substitution p. 141 (13, 14, 15, 16)Solve by elimination p. 142 (50, 52, 54, 56)Solve by any method p. 142 (29, 49, 70,71,76)