3.2 Solving Systems Algebraically*Substitution*Elimination

Solving using Substitution Solve for one of the variables

in one of the equations with a coefficient of 1

Substitute the expression into the other equation

Solve the equation Plug answer back into the original

equation to find the other variable Write your two answers together as an

ordered pair

Examples 4x + 3y = 4 2x – y + 7

2x – 3y = 6 x + y = -12

Practice!!1) 3x – y = 0 4x + 3y = 26

2) 2x + 3y = 12 -2x + 4y = 9

Warm-up Solve by Substitution

1. y = x + 4 y = 3x

2. y = x – 7 2x + y = 8

( 1, 1 ) ( 5 , -2 )

Solving using Elimination Get one set of variables coefficients to be

equal but opposite (ex. 3 and -3) Using multiplication

Add the two equations together vertically Solve for the remaining variable Plug answer back into the original

equation to get the other variable Write your two answers together as an

ordered pair

What if the variables both cancel out? If both variables are eliminated -and you get a true statement they are

the same line and there are infinite solutions.

-and you get a false statement the lines are parallel and there are no solutions.

Mike and Caryn bought some pens and pencils. Mike bought 4 pens and 5 pencils, which cost him $6.71. Caryn bought 5 pens and 3 pencils, which cost her $7.12. Write a system of equations and solve to determine the price of each pen and pencil.

You have just enough money to buy a loaf of bread for $1.95. You have 12 coins, all quarters and dimes. Write and solve a system of equations to determine how many of each type of coin you have.

Revenue – Expenses = Profit

Suppose you are starting an officecleaning service. You have spent $315 on equipment. To clean each office, you use $4 worth of supplies. You charge $25 per office. How manyOffices must you clean to break even?

15 offices

Identify variables! Write and solve a system

Suppose you bought supplies for a party. Three rolls of streamers and 15 party hats cost $30. Later, you bought 2 rolls of streamers and 4 party hats for $11. How much did each roll of streamers cost? How much did each party hat cost?

streamers$2.50/hats$1.50

Examples4x – 2y = 7x + 2y = 3

4x + 9y = 14x + 6y = -2

2x + 4y = -43x + 5y = -3

2x – y = 3-2x + y = -3

2x – 3y = 18-2x + 3y = -6

Practice!!!!1) -3x + 5y = 6 6x – 10y = 0

2) 3x + y = -9 -3x – 2y = 12

3) -3x + 5y = 7 6x – 10y = -14

4) 5x – 2y = -19 2x + 3y = 0

Warm up Solve the following systems by elimination.

( 2, -1 ) ( 2, 0 ) (1/2, 2 )

1232

432663

02152

yxyx

yxyx

yxyx

Three pizzas and four sandwiches cost $34. Three pizzas and seven sandwiches cost $41.50. How much does a pizza cost? Identify variables, write and solve a system of equations.