Algebra-2Algebra-2

Section 3-2BSection 3-2B

Quiz 3-2A:Quiz 3-2A:Solve using substitution.Solve using substitution.

2x + y = -22x + y = -2-2x + 3y = -8-2x + 3y = -8

11. .

22. . 3x – 4y = -103x – 4y = -106x + 3y = -426x + 3y = -42

VocabularyVocabulary

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

VocabularyVocabularyElimination MethodElimination Method: Eliminate one of the: Eliminate one of thevariables by adding the equations together.variables by adding the equations together.

x – 3y = 5x – 3y = 5-x + 5y = 3-x + 5y = 3

What property allows me toWhat property allows me to add equations together? add equations together?

““same thing left, same thing right” same thing left, same thing right”

Adding these equations will Adding these equations will eliminateeliminate the ‘x’ variable. the ‘x’ variable.

2x – 3y = 52x – 3y = 5-4x + 3y = 3-4x + 3y = 3

Adding these equations will Adding these equations will eliminateeliminate the ‘y’ variable. the ‘y’ variable.

Your turn:Your turn:What variable will be eliminated if I add What variable will be eliminated if I add

the following equations? the following equations?

2x + y = -22x + y = -2-2x + 3y = -8-2x + 3y = -8

11. .

22. . 4x – 3 y = -24x – 3 y = -2-2x + 3y = -8-2x + 3y = -8

33. . 3x + y = -13x + y = -12x + 3y = 182x + 3y = 18

ExampleExampleEliminate one of the variables by adding Eliminate one of the variables by adding the equations together.the equations together.

x – 3y = 5x – 3y = 5-x + 5y = 3-x + 5y = 3

x – x – 3y + 5y = 5 + 3x – x – 3y + 5y = 5 + 3

2y = 82y = 8y = 4y = 4

Replace ‘y’ with 4 in Replace ‘y’ with 4 in either of the original either of the original equations, then solveequations, then solve for ‘x’.for ‘x’.

x – 3(4) = 5x – 3(4) = 5x = 17x = 17 Solution: (17, 4)Solution: (17, 4)

Check the solution:Check the solution:

x – 3y = 5x – 3y = 5-x + 5y = 3-x + 5y = 3

Replace ‘x’ with 17 and ‘y’ with 4 in Replace ‘x’ with 17 and ‘y’ with 4 in bothboth of the original of the original equations, to see if the ordered pair (17, 4) is a solution equations, to see if the ordered pair (17, 4) is a solution to the system of equations. to the system of equations.

Checks!Checks!

Solution: (17, 4)Solution: (17, 4)

(17) – 3(4) = 5(17) – 3(4) = 5 -(17) + 5(4) = 3-(17) + 5(4) = 3Checks!Checks!

VocabularyVocabularyElimination MethodElimination Method: Eliminate one of the: Eliminate one of thevariables by adding the equations together.variables by adding the equations together.

2x – 5y = 62x – 5y = 6-x + 5y = 2-x + 5y = 2

2x – x – 5y + 5y = 6 + 22x – x – 5y + 5y = 6 + 2

x = 8x = 8 Replace ‘x’ with 8 in Replace ‘x’ with 8 in either of the original either of the original equations, then solveequations, then solve for ‘y’.for ‘y’.

-(8) + 5y = 2-(8) + 5y = 2

5y = 105y = 10Solution: (8, 2)Solution: (8, 2)

y = 2y = 2

Your turn:Your turn:Solve the equation using “elimination” Solve the equation using “elimination”

2x – y = -22x – y = -2-2x + 3y = -8-2x + 3y = -8

4. 4.

102 y5y

2)5(2 x

72 x

2

7x

252 x

Your turn:Your turn:Solve the equation using “elimination” Solve the equation using “elimination”

5. 5. 4x – 3 y = -24x – 3 y = -2-2x + 3y = -8-2x + 3y = -8

102 x

5y

2)5(32 x2152 x

172 x

2

7x

Steps to take to make the Steps to take to make the equationsequations “addable” (so you can eliminate a “addable” (so you can eliminate a variable).variable).Can you add the two equations together to Can you add the two equations together to

eliminate the one of the variables?eliminate the one of the variables?

5x – y = -25x – y = -2-2x + 3y = -8-2x + 3y = -8

(3)(3)5x – 5x – (3)(3)y = -2y = -2(3)(3) -2x + 3y = -8-2x + 3y = -8

1515x – x – 33y = -6y = -6 -2x + 3y = -8-2x + 3y = -8

Steps to take to make the Steps to take to make the equationsequations “addable” (so you can eliminate a “addable” (so you can eliminate a variable).variable).Can you add the two equations together to Can you add the two equations together to

eliminate the one of the variables?eliminate the one of the variables?

4x + 2y = -14x + 2y = -12x + 3y = 182x + 3y = 18

4x + 2y = -14x + 2y = -1(-2)(-2)2x + 2x + (-2)(-2)3y = 183y = 18(-2)(-2)

4x + 2y = -14x + 2y = -1-4-4x x – 6– 6y = y = -36-36

2x – 2y = 62x – 2y = 6-x + 6y = 7-x + 6y = 7

EliminationElimination

What happens if just by adding two equationsWhat happens if just by adding two equationsOne of the variables doesn’t disappear? One of the variables doesn’t disappear?

2x – 2y = 62x – 2y = 6 -x + 6y = 7-x + 6y = 7

Use properties of Use properties of EqualityEquality: multiply one: multiply oneOf the equations by aOf the equations by aNumber so that the Number so that the leadleadCoefficientCoefficient is the is the samesameNumber but opposite signNumber but opposite signAs the other equation.As the other equation.

(2)(2) (2)(2)

2x – 2y = 62x – 2y = 6 -2-2x + x + 1212y = y = 1414

10y = 2010y = 20

y = 2y = 2 x = ?x = ?

Your turn:Your turn:Solve the equations using “elimination” Solve the equations using “elimination”

6. 6. 3x + y = -13x + y = -12x + 3y = 182x + 3y = 18

(-3)(-3)3x + 3x + (-3)(-3)y = -1y = -1(-3)(-3) 2x + 3y = 182x + 3y = 18

-9-9x x – 3– 3y = y = 33 2x + 3y =182x + 3y =18

217 x

3x

183)3(2 y

1836 y123 y4y

Your turn:Your turn:Solve using the elimination method:Solve using the elimination method:

7.7. 262 yx

9102 yx

2x + 6y = 22x + 6y = 2(-1)(-1)2x – 2x – (-1)(-1)10y = 910y = 9(-2)(-2)

2x + 6y = 22x + 6y = 2-2-2x + x + 1010y = y = -18-18

1616 y1y

9)1(102 x9102 x12 x

2

1x

Your turn:Your turn:Solve using the elimination method:Solve using the elimination method:

8.8. 1043 yx4236 yx

(-2)(-2)3x – 3x – (-2)(-2)4y = -104y = -10(-2)(-2) 6x + 3y = -426x + 3y = -42

-6-6x + x + 88y = y = 2020 6x + 3y = -426x + 3y = -42

2211 y2y

42)2(36 x

4266 x366 x

6x

Your Your turn:turn:

3x + 2y = 63x + 2y = 6 x – 4y = -12x – 4y = -12

(2)(2)3x + 3x + (2)(2)2y = 62y = 6(2)(2) x – 4y = -12x – 4y = -12

66x + x + 44y = y = 1212 x – 4y = -12x – 4y = -12

5x = 05x = 0

x = 0x = 0

3 3 (0) (0) + 2y = 6+ 2y = 6 (0)(0) – 4y = -12 – 4y = -12

2y = 62y = 6-4y = -12-4y = -12

y = 3y = 3

Solution is Solution is (0, 3)(0, 3)

9.9.

What do you do for this What do you do for this case?case? 2x + 5y = 142x + 5y = 14 3x – 2y = -363x – 2y = -36

(3)(3)2x + 2x + (3)(3)5y = 145y = 14(3)(3)(-2)(-2)3x – 3x – (-2)(-2)2y2y = -36 = -36 (-2)(-2)

Multiply Multiply bothboth equations equations to get a to get a common coefficientcommon coefficient..

66x + 15y = 42x + 15y = 42 -6-6x + 4y = 72x + 4y = 72

19y = 11419y = 114

y = 6y = 6

2x + 52x + 5(6)(6) = 14 = 14 3x – 23x – 2(6)(6) = -36 = -36

2x + 2x + 3030 = 14 = 14 3x – 3x – 1212 = -36 = -36

2x = 2x = -16-16 3x = 3x = -24-24

x = -8x = -8Solution is Solution is (-8, 6)(-8, 6)

Your turn:Your turn: Solve using the elimination method:Solve using the elimination method:

10.10. 1127 yx

2932 yx

Categories of Solutions:Categories of Solutions:

Ways 2 lines can be graphed:Ways 2 lines can be graphed:

Cross Cross one solution one solution

Parallel Parallel no solutionsno solutions

Same line Same line infinite infinite number of number of solutionssolutions

How do you know?How do you know? (1, 0, or infinite #) (1, 0, or infinite #)Use elimination and Use elimination and both variablesboth variables disappear and disappear and the resulting equation is either: the resulting equation is either:

a. a. truetrue:: ( 3 = 3 or 0 = 0)( 3 = 3 or 0 = 0) Infinite # of solutionsInfinite # of solutions

b. b. falsefalse:: ( -2 = 3 or 10 = 0)( -2 = 3 or 10 = 0) No solutionsNo solutions

Your turn:Your turn:1111. Solve using the . Solve using the elimination methodelimination method::

2x + y = -22x + y = -26x + 3y = -86x + 3y = -8