Solving a System of Equations using Multiplication

Chapter 6-4

Solving Systems using MultiplicationNCSCOS 4.03

Students will know how to solve a system of equations by using multiplication

Solving Systems using MultiplicationLet’s look at a simple

example:

Can I multiply both sides?Yes!

As long as I do the same thing on both sides, it will still be equal!

Notice that the numbers are bigger, but it’s still equal

3 = 32 26 = 6

Key Concept:

1.

Solving Systems using MultiplicationAre we allowed to

multiply an equation like the one on the right?

As long as we do the same thing on the right as we do on the left we can!

If we multiply both sides by 2 it will still be the same equation but it changes the numbers

2x + 3y = 44x + 6y = 82

Key Concept:

2.

3.

Solving Systems using MultiplicationExample 1: Find the solution to the following

system of equations: x + y = -2 and 2x + y = 6.

In order to solve a system of equations we have to eliminate one of the variables

First look to see if you can add the equations to eliminate a variable

If you can’t, you’re going to have to multiply one of the equations first

4.

5.

6.

Solving Systems using MultiplicationLet’s multiply the

first equation by -1Remember to

multiply all the numbers by the -1

The new equation is still the same as the first, but the numbers are different

We can now use addition to solve this system

2x + y = 6x + y = -2-1

-x – y = 27.

8.

9.

Solving Systems using MultiplicationAdd the two equations

to eliminate the y’s

Now that we know what x is, we can plug it into either equation to solve for y

It doesn’t matter which equation we use

-x – y = 22x + y = 6+________

x = 810.

11.

Solving Systems using MultiplicationPlug 8 in for x

Solve for y

Now that we know y and x we can put them together for our answer!

That’s the point where the two lines intersect

x + y = -28 + y = -2-8 -8

y = -10

(8, -10)12.13.

Solving Systems using MultiplicationSolve the following system of equations using

multiplication.

1. x + y = -3x + 2y = 13. 3x – 5y = -352x – 5y = -30

5. 4x + 5y = 78x + 5y = 9

4. 13x + 5y = -1113x + 11y = 7

2. 3x + y = 5

2x + y = 10

Practice 1

Solving Systems using MultiplicationSolve the following system of equations using

multiplication.

1. x + y = -3x + 2y = 13. 3x – 5y = -352x – 5y = -30

5. 4x + 5y = 78x + 5y = 9

4. 13x + 5y = -1113x + 11y = 7

2. 3x + y = 5

2x + y = 10

(-7, 4)

(1/2, 1)

(-2, 3)(-5, 4)

(-5, 20)

Practice 1

Solving Systems using MultiplicationExample 2: Use elimination to solve the

system of equations.

3x + 4y = 6-5x – 2y = 4

First we have to see if we can add them together to eliminate one of the variablesNO!

So we’ll have to find a different way to solve

14.

Solving Systems using MultiplicationTo eliminate a

variable they must add up to zero

One has to be negative and one has to be positive

Since the y values are opposite signs and 4 is a multiple of 2 we will eliminate the y values first

3x + 4y = 6-5x – 2y = 4

17.

16.

15.

Solving Systems using MultiplicationIn order to get rid

of the y values we have to multiply the first equation by 2

This doesn’t change the actual equation, just the numbers

3x + 4y = 6-5x – 2y = 42

-10x – 4y = 8

Solving Systems using MultiplicationNow we can add

the equations together to eliminate the y’s

Divide by -7

Now that we know x, plug it back into any equation to find y

3x + 4y = 6-10x – 4y = 8+___________-7x = 14___ __-7 -7

x = -219.

Solving Systems using MultiplicationIt doesn’t matter

which equation we use, so I’ll plug x into the second equation because the number are positive

Now solve for yMultiply on the leftAdd 6 to both sidesDivide by 4

3x + 4y = 63(-2) + 4y = 6

-6 + 4y = 6+6 +6

4y = 12__ __4 4

20.

Solving Systems using MultiplicationNow that we have y, plug

the values of x and y into a point for your answer!

y = 3(-2, 3)

22.

Solving Systems using MultiplicationUse elimination to solve each system of

equations

1. -5x + 3y = 6 x – y = 4

3. 2x + y = 53x – 2y = 4

5. 5x – 2y = -153x + 8y = 37

4. 4x – 3y = 12 x + 2y = 14

2. x + y = 32x – 3y = 16

Practice 2

Solving Systems using MultiplicationUse elimination to solve each system of

equations

1. -5x + 3y = 6 x – y = 4

3. 2x + y = 53x – 2y = 4

5. 5x – 2y = -153x + 8y = 37

4. 4x – 3y = 12 x + 2y = 14

2. x + y = 32x – 3y = 16

(-9, -13)

(6, 4)(2, 1)

(5, -2)

(-1, 5)Practice 2

Solving Systems using MultiplicationExample 3: Solve the following system of

equations:First check to see if

you can just add these equations together to eliminate a variable

Since we can’t do that, can we multiply one equation to eliminate a variable?No!

3x + 4y = -252x – 3y = 623.

24.

Solving Systems using MultiplicationExample 3: Solve the following system of

equations:What we’ll need to

do is multiply both equations by a different number so that we can cancel out a variable

3x + 4y = -252x – 3y = 6

I would look at eliminating the y values since they have opposite signs already

25.

26.

Solving Systems using MultiplicationMultiply the first

equation by 3Multiply the second

equation by 4

Now we can add these equations together to eliminate the y’s!

3x + 4y = -252x – 3y = 634

9x + 12y = -758x – 12y = 2428.

Solving Systems using MultiplicationAdd the equations

Divide by 17

Plug x into either of the original equations

9x + 12y = -758x – 12y = 24+___________

17x = -51___ __17 17

x = -3

Solving Systems using MultiplicationIt doesn’t matter if

you use either of the original equations or the ones that you multiplied, but I will use the second equation since the numbers are smaller

Solve for yAdd 6 Divide by -3

2x – 3y = 62(-3) – 3y = 6

-6 – 3y = 6+6 +6

– 3y = 12___ __-3 -3

Solving Systems using MultiplicationPlug your x and y values

into a point for your answer

y = -4

(-3, -4)31.

Solving Systems using MultiplicationSolve the following system of equations

1. 5x – 2y = -15 3x + 8y = 37

3. 2x – 3y = 2 5x + 4y = 28

4. 4x + 3y = 19 3x – 4y = 8

5. 4x + 7y = 6 6x + 5y = 20

2. 4x – 7y = 10 3x + 2y = -7

Practice 3

Solving Systems using MultiplicationSolve the following system of equations

1. 5x – 2y = -15 3x + 8y = 37

3. 2x – 3y = 2 5x + 4y = 28

4. 4x + 3y = 19 3x – 4y = 8

5. 4x + 7y = 6 6x + 5y = 20

2. 4x – 7y = 10 3x + 2y = -7(-1,

5)

(4, 1)(4, 2)

(5, -2)

(-1, -2)

Practice 3

Solving Systems using MultiplicationSolve the following system of equations

1. x – y = -5 3x – y = 7

3. 5x + 2y = 7 3x – y = 13

2. 3x – 3y = 6 3x + y = -10

4. -5x + 2y = -18

3x + 4y = -105. 4x + 3y = -11

3x + 5y = 0

Quiz!

Solving Systems using MultiplicationSolve the following system of equations

1. x – y = -5 3x – y = 7

3. 5x + 2y = 7 3x – y = 13

2. 3x – 3y = 6 3x + y = -10

4. -5x + 2y = -18

3x + 4y = -105. 4x + 3y = -11

3x + 5y = 0

(6, 11)

(2, -4)

(-2, -4)

(3, -4)

(-5, 3)

Quiz!