Algebra-2 Section 3-2B

Quiz 3-2A: Solve using substitution. 1. 2x + y = -2 -2x + 3y = -8 2.

Vocabulary Elimination Method: Eliminate one of the variables by adding the two equations together.

Vocabulary x – 3y = 5 -x + 5y = 3 2x – 3y = 5 -4x + 3y = 3 Elimination Method: Eliminate one of the variables by adding the equations together. What property allows me to add equations together? x – 3y = 5 -x + 5y = 3 “same thing left, same thing right” Adding these equations will eliminate the ‘x’ variable. 2x – 3y = 5 -4x + 3y = 3 Adding these equations will eliminate the ‘y’ variable.

Your turn: What variable will be eliminated if I add the following equations? 1. 2x + y = -2 -2x + 3y = -8 2. 4x – 3 y = -2 -2x + 3y = -8 3. 3x + y = -1 2x + 3y = 18

Example x – 3y = 5 -x + 5y = 3 x – x – 3y + 5y = 5 + 3 2y = 8 y = 4 Eliminate one of the variables by adding the equations together. x – 3y = 5 -x + 5y = 3 x – x – 3y + 5y = 5 + 3 Replace ‘y’ with 4 in either of the original equations, then solve for ‘x’. 2y = 8 y = 4 x – 3(4) = 5 x = 17 Solution: (17, 4)

Check the solution: -(17) + 5(4) = 3 (17) – 3(4) = 5 x – 3y = 5 Replace ‘x’ with 17 and ‘y’ with 4 in both of the original equations, to see if the ordered pair (17, 4) is a solution to the system of equations. x – 3y = 5 -x + 5y = 3 -(17) + 5(4) = 3 (17) – 3(4) = 5 Checks! Checks! Solution: (17, 4)

Vocabulary 2x – 5y = 6 -x + 5y = 2 2x – x – 5y + 5y = 6 + 2 x = 8 Elimination Method: Eliminate one of the variables by adding the equations together. 2x – 5y = 6 -x + 5y = 2 2x – x – 5y + 5y = 6 + 2 Replace ‘x’ with 8 in either of the original equations, then solve for ‘y’. x = 8 -(8) + 5y = 2 5y = 10 Solution: (8, 2) y = 2

Your turn: 2x – y = -2 -2x + 3y = -8 4. Solve the equation using “elimination” 2x – y = -2 -2x + 3y = -8 4.

Your turn: 4x – 3 y = -2 -2x + 3y = -8 5. Solve the equation using “elimination” 4x – 3 y = -2 -2x + 3y = -8 5.

Steps to take to make the equations “addable” (so you can eliminate a variable). Can you add the two equations together to eliminate the one of the variables? 5x – y = -2 -2x + 3y = -8 (3)5x – (3)y = -2(3) -2x + 3y = -8 15x – 3y = -6 -2x + 3y = -8

Steps to take to make the equations “addable” (so you can eliminate a variable). Can you add the two equations together to eliminate the one of the variables? 4x + 2y = -1 2x + 3y = 18 4x + 2y = -1 (-2)2x + (-2)3y = 18(-2) 4x + 2y = -1 -4x – 6y = -36

Elimination 2x – 2y = 6 -x + 6y = 7 2x – 2y = 6 -x + 6y = 7 What happens if just by adding two equations One of the variables doesn’t disappear? Use properties of Equality: multiply one Of the equations by a Number so that the lead Coefficient is the same Number but opposite sign As the other equation. 2x – 2y = 6 -x + 6y = 7 (2) (2) 2x – 2y = 6 -2x + 12y = 14 10y = 20 y = 2 x = ?

Your turn: 3x + y = -1 (-3)3x + (-3)y = -1(-3) 6. 2x + 3y = 18 Solve the equations using “elimination” 6. 3x + y = -1 2x + 3y = 18 (-3)3x + (-3)y = -1(-3) 2x + 3y = 18 -9x – 3y = 3 2x + 3y =18

Your turn: 7. 2x + 6y = 2 (-1)2x – (-1)10y = 9(-2) 2x + 6y = 2 Solve using the elimination method: 7. 2x + 6y = 2 (-1)2x – (-1)10y = 9(-2) 2x + 6y = 2 -2x + 10y = -18

Your turn: 8. (-2)3x – (-2)4y = -10(-2) 6x + 3y = -42 -6x + 8y = 20 Solve using the elimination method: 8. (-2)3x – (-2)4y = -10(-2) 6x + 3y = -42 -6x + 8y = 20 6x + 3y = -42

Your turn: 9. 3x + 2y = 6 x – 4y = -12 3 (0) + 2y = 6 (0) – 4y = -12 Solution is (0, 3) 5x = 0 x = 0

What do you do for this case? 2x + 5y = 14 3x – 2y = -36 Multiply both equations to get a common coefficient. (3)2x + (3)5y = 14(3) (-2)3x – (-2)2y = -36 (-2) 2x + 5(6) = 14 3x – 2(6) = -36 6x + 15y = 42 -6x + 4y = 72 2x + 30 = 14 3x – 12 = -36 2x = -16 3x = -24 19y = 114 y = 6 x = -8 Solution is (-8, 6)

Your turn: Solve using the elimination method: 10.

Categories of Solutions: Ways 2 lines can be graphed: Cross  one solution Parallel no solutions Same line  infinite number of solutions

How do you know? (1, 0, or infinite #) Use elimination and both variables disappear and the resulting equation is either: a. true: ( 3 = 3 or 0 = 0) Infinite # of solutions b. false: ( -2 = 3 or 10 = 0) No solutions

Your turn: 11. Solve using the elimination method: 2x + y = -2 6x + 3y = -8