Title:  Elimination Using Multiplication
Grade: 9
Section: 7.4

Materials:  calculators, worksheets, chalkboard/whiteboard, chalk/markers
**ALL WORKSHEETS WILL BE PROVIDED

Lesson Overview:  Students will learn how to solve system of equations by using elimination with multiplication.  They will also determine the best method for solving systems of equations.

Lesson Objectives:  
•	Students will be able to analyze the system and determine the best method to solve it.  [Analysis]
•	Students will analyze and solve problems whose solution requires solving systems of linear equations in two variables, by multiplication.  [Analysis]
•	Given a word problem the students will be able to create a system of equations from the information, solve through elimination (multiplication), and then evaluate the appropriateness of their answer.  [Evaluation]

NYS Standards:
•	Key Idea 3: Operations- Use addition, subtraction, multiplication, division, and exponentiation with real numbers and algebraic expressions.
•	Key Idea 4:  Modeling/Multiple Representation- Model real-world problems with systems of equations and inequalities.
•	Key Idea 7:  Patterns/Functions- Represent and analyze functions, using verbal descriptions and equations.

Anticipatory Set: (5 mins)
Pass out “Rewind for a Moment” worksheet.  Students should complete this independently, and then go over as a class.  This is a brief practice of least common multiples and scale factor.  This will hopefully get the students to think about multiplying to get two numbers to be equal.
 
Developmental Activity: (25 mins)
1.)	Pass out the Elimination Using Multiplication worksheet.
2.)	Complete the worksheet as a whole group.
3.)	The examples are broken up into steps, with questions to help guide the students thought process.  Be sure to discuss the students’ responses.

Closure: (5 mins)
Discuss what they learned about elimination.  When do you use multiplication?  Which variable do you find first?  Why do we use multiplication before adding or subtracting?  These are just a few suggestions.

Assessment:  (5 mins)
Have the students complete the Exit Ticket and hand it in before they leave.  This will be graded.

Name:									Date:

Rewind for a Moment

1.)  

           	3	              3                      6                               6

                          4                                                8

What number were the sides of the smaller triangle multiplied by in order to get the side lengths of the larger triangle?

2.)  You have the equation 3x + 10 = 37, what will you have to multiple it by in order to          make that equation equal to 9x + 30 = 111?

Name:	ANSWER KEY							Date:

Rewind for a Moment

1.)  

           	3	              3                      6                               6

                          4                                                8

What number were the sides of the smaller triangle multiplied by in order to get the side lengths of the larger triangle?

		2

2.)  You have the equation 3x + 10 = 37, what will you have to multiple it by in order to make that equation equal to 9x + 30 = 111?

			3

Name:	ANSWER KEY								Date:

Elimination Using Multiplication

Sometimes neither variable in a system of equations can be eliminated by simply adding or subtracting the equations.  However, you can use the MULTIPLICATION PROPERTY of EQUALITY so that adding or subtracting eliminates one of the variables.

1.)	Multiply One Equation to Eliminate:
3x + 4y = 6
	5x + 2y = -4

Let’s try to eliminate “y” first.
How can I get the y coefficients to be the same number with different signs?
MULTIPLY 5x + 2y = -4 BY -2.

Why did you choose that number?
BECAUSE 2y * -2 = -4y, WHICH IS WHAT WE NEED IT TO BE IN ORDER TO ELIMINATE y.

Solve for x:
      3x + 4y = 6
		(-2) *(5x + 2y = -4)
		   -10x – 4y = 8

         3x + 4y = 6
		(+) -10x – 4y = 8
		        -7x         = 14
			x = -2
				
How will we get our y-value? PLUG IN -2 FOR X INTO ONE OF THE EQUATIONS

Solve for y:    3(-2) +4y = 6
		          4y =  12
			y = 3

The solution of the system is? (-2, 3)
Check answer:  3(-2) + 4(3) = 6
		  5(-2) + 2(3) = -4

2.)	Multiply Both Equations to Eliminate:
3x + 4y = -25
2x – 3y = 6

Method 1:  Eliminate x first.
3x + 4y = -25    MULTIPLY BY 2			6x + 8y = -50
2x – 3y = 6	  MULTIPLY BY -3		   (+)  -6x + 9y = -18
								      17y = -68
								          y = -4

How do we get our x-value? PLUG IN -4 FOR y
Solve for x:
	3x + 4(-4) = -25
	3x = -9
	x = -3
The solution of the system is? (-3, -4)
Check Answer:  3(-3)+4(-4)=25	         2(-3) – 3(-4)= 6

Method 2:  Eliminate y first.
	3x + 4y = -25    MULTIPLY BY 3			 9x + 12y  = -75
2x – 3y = 6	  MULTIPLY BY 4		       (+)  8x – 12y  = 24
							17x	    = -51
								x= -3
How do we get our y-value? PLUG IN -3 FOR x
Solve for y:
	2(-3) +- 3y = 6
		-3y = 12
		   y =  -4
The solution of the system is? (-3, -4)
Check Answer:  3(-3)+4(-4)=25	         2(-3) – 3(-4)= 6

Over the past few classes you have learned a variety of ways to solve systems of equations.  Fill in the blanks to complete the chart on solving systems of equations.

METHOD	THE BEST TIME TO USE	
GRAPHING	to estimate the solution, since graphing usually does not give an exact solution	
Substitution	if one of the variables in either equation has a coefficient of 1 or -1	
Elimination Using ADDITION	if one of the variables has opposite coefficients in the two equations	
Elimination Using Subtraction	if one of the variables has the SAME coefficient in the two equations	
ELIMINATION Using Multiplication	if none of the coefficients are 1 or -1 and neither of the variables can be eliminated by simply adding or SUBTRACTING the equations	

Name:											Date:

Elimination Using Multiplication

Sometimes neither variable in a system of equations can be eliminated by simply adding or subtracting the equations.  However, you can use the MULTIPLICATION PROPERTY of EQUALITY so that adding or subtracting eliminates one of the variables.

1.)	 Multiply One Equation to Eliminate:
3x + 4y = 6
	5x + 2y = -4

Let’s try to eliminate “y” first.
How can I get the y coefficients to be the same number with different signs?
________________________________________________________________________

Why did you choose that number?
________________________________________________________________________

Solve for x:

How will we get our y-value? _______________________________________________

Solve for y:    

The solution of the system is? ________
Check answer: 

2.)  Multiply Both Equations to Eliminate:
3x + 4y = -25
2x – 3y = 6

Method 1:  Eliminate x first.
3x + 4y = -25    	MULTIPLY BY ____
2x – 3y = 6	  	MULTIPLY BY ____
How do we get our x-value? ________________________________________________
Solve for x:

The solution of the system is? _________
Check Answer: 

Method 2:  Eliminate y first.
	3x + 4y = -25    MULTIPLY BY ____			
2x – 3y = 6	  MULTIPLY BY ____		       

How do we get our y-value? ________________________________________________
Solve for y:

The solution of the system is? ________
Check Answer: 

Over the past few classes you have learned a variety of ways to solve systems of equations.  Fill in the blanks to complete the chart on solving systems of equations.

METHOD	THE BEST TIME TO USE	
_______________________	to estimate the solution, since graphing usually does not give an exact solution	
Substitution	if one of the variables in either equation has a coefficient of 1 or -1	
Elimination Using ____________	if one of the variables has opposite coefficients in the two equations	
Elimination Using Subtraction	if one of the variables has the __________ coefficient in the two equations	
_________________ Using Multiplication	if none of the coefficients are 1 or -1 and neither of the variables can be eliminated by simply adding or _______________ the equations	

Name:										Date:
Exit Ticket

1.)  What would be the best method to use to solve the following system of equations?
	3x – 4y = -10
	5x + 8y = -2

2.)  Use elimination to solve the system of equations.
	x + 5y = 4
          3x  - 7y = -10	

Name:	ANSWER KEY								Date:
Exit Ticket

1.)  What would be the best method to use to solve the following system of equations?
	3x – 4y = -10
	5x + 8y = -2
				SUBSTITUTION

2.)  Use elimination to solve the system of equations.
	x + 5y = 4	MULTIPLY BY -3		  -3x – 15y = -12
          3x  - 7y = -10				       (+)	    3x  - 7y = -10
							         -22y  =  -22
								y = 1
								x + 5(1)= 4
								x = -1
								(-1,1)
							-1 + 5 (1) = 4
							3(-1) – 7(1) = -10