7.3

Solving Systems Using

Elimination

7.3 – Solving by Elimination

Goals / “I can…”Solve systems by adding or subtractingMultiply first when solving systems

Solving Systems of Equations

So far, we have solved systems using graphing, substitution, and elimination. These notes go one step further and show how to use ELIMINATION with multiplication.

What happens when the coefficients are not the same?

We multiply the equations to make them the same! You’ll see…

Elimination using Addition

Consider the system

x - 2y = 5

2x + 2y = 7

REMEMBER: We are trying to find the Point of Intersection. (x, y)

Lets add both equations to each other

Elimination using Addition

Consider the system

x - 2y = 5

2x + 2y = 7

Lets add both equations to each other+

NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition

Consider the system

x - 2y = 5

2x + 2y = 7

Lets add both equations to each other+

3x = 12x = 4

ANS: (4, y)

NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition

Consider the system

x - 2y = 5

2x + 2y = 7

ANS: (4, y)

Lets substitute x = 4 into this equation.

4 - 2y = 5 Solve for y - 2y = 1

y = 12

NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition

Consider the system

x - 2y = 5

2x + 2y = 7

ANS: (4, )

Lets substitute x = 4 into this equation.

4 - 2y = 5 Solve for y - 2y = 1

y = 12

12

NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition

Consider the system

3x + y = 14

4x - y = 7

NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition

Consider the system

3x + y = 14

4x - y = 7

7x = 21x = 3

ANS: (3, y)

+

Elimination using Addition

Consider the system

ANS: (3, )

3x + y = 14

4x - y = 7

Substitute x = 3 into this equation

3(3) + y = 149 + y = 14

y = 5

5

NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Multiplication

Consider the system

6x + 11y = -5

6x + 9y = -3

Elimination using Multiplication

Consider the system

6x + 11y = -5

6x + 9y = -3+12x + 20y = -8 When we add equations together,

nothing cancels out

Elimination using Multiplication

Consider the system

6x + 11y = -5

6x + 9y = -3

Elimination using Multiplication

Consider the system

6x + 11y = -5

6x + 9y = -3

-1 ( )

Elimination using Multiplication

Consider the system

- 6x - 11y = 5

6x + 9y = -3+-2y = 2

y = -1

ANS: (x, )-1

Elimination using Multiplication

Consider the system

6x + 11y = -5

6x + 9y = -3

ANS: (x, )-1

y = -1

Lets substitute y = -1 into this equation

6x + 9(-1) = -36x + -9 = -3

+9 +9

6x = 6x = 1

Elimination using Multiplication

Consider the system

6x + 11y = -5

6x + 9y = -3

ANS: ( , )-1

y = -1

Lets substitute y = -1 into this equation

6x + 9(-1) = -36x + -9 = -3

+9 +9

6x = 6x = 1 1

Solving a system of equations by elimination using multiplication.

Step 1: Put the equations in Standard Form.

Step 2: Determine which variable to eliminate.

Step 3: Multiply the equations and solve.

Step 4: Plug back in to find the other variable.

Step 5: Check your solution.

Standard Form: Ax + By = C

Look for variables that have the

same coefficient.

Solve for the variable.

Substitute the value of the variable

into the equation.

Substitute your ordered pair into

BOTH equations.

1) Solve the system using elimination.

2x + 2y = 6

3x – y = 5Step 1: Put the equations in

Standard Form.

Step 2: Determine which variable to eliminate.

They already are!

None of the coefficients are the same!

Find the least common multiple of each variable.

LCM = 6x, LCM = 2y

Which is easier to obtain?

2y(you only have to multiplythe bottom equation by 2)

1) Solve the system using elimination.

Step 4: Plug back in to find the other variable.

2(2) + 2y = 6

4 + 2y = 6

2y = 2

y = 1

2x + 2y = 6

3x – y = 5

Step 3: Multiply the equations and solve.

Multiply the bottom equation by 2

2x + 2y = 6

(2)(3x – y = 5)

8x = 16

x = 2

2x + 2y = 6(+) 6x – 2y = 10

1) Solve the system using elimination.

Step 5: Check your solution.

(2, 1)

2(2) + 2(1) = 6

3(2) - (1) = 5

2x + 2y = 6

3x – y = 5

Solving with multiplication adds one more step to the elimination process.

2) Solve the system using elimination.

x + 4y = 7

4x – 3y = 9Step 1: Put the equations in

Standard Form.They already are!

Step 2: Determine which variable to eliminate.

Find the least common multiple of each variable.

LCM = 4x, LCM = 12y

Which is easier to obtain?

4x(you only have to multiplythe top equation by -4 to

make them inverses)

2) Solve the system using elimination.

x + 4y = 7

4x – 3y = 9

Step 4: Plug back in to find the other variable.

x + 4(1) = 7

x + 4 = 7

x = 3

Step 3: Multiply the equations and solve.

Multiply the top equation by -4

(-4)(x + 4y = 7)

4x – 3y = 9)

y = 1

-4x – 16y = -28 (+) 4x – 3y = 9

-19y = -19

2) Solve the system using elimination.

Step 5: Check your solution.

(3, 1)

(3) + 4(1) = 7

4(3) - 3(1) = 9

x + 4y = 7

4x – 3y = 9

What is the first step when solving with elimination?

1. Add or subtract the equations.

2. Multiply the equations.

3. Plug numbers into the equation.

4. Solve for a variable.

5. Check your answer.

6. Determine which variable to eliminate.

7. Put the equations in standard form.

Which variable is easier to eliminate?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32

3x + y = 44x + 4y = 6

1. x

2. y

3. 6

4. 4

3) Solve the system using elimination.

3x + 4y = -1

4x – 3y = 7

Step 1: Put the equations in Standard Form.

They already are!

Step 2: Determine which variable to eliminate.

Find the least common multiple of each variable.

LCM = 12x, LCM = 12y

Which is easier to obtain?

Either! I’ll pick y because the signs are already opposite.

3) Solve the system using elimination.

3x + 4y = -1

4x – 3y = 7

Step 4: Plug back in to find the other variable.

3(1) + 4y = -1

3 + 4y = -1

4y = -4

y = -1

Step 3: Multiply the equations and solve.

Multiply both equations

(3)(3x + 4y = -1)

(4)(4x – 3y = 7)

x = 1

9x + 12y = -3 (+) 16x – 12y = 28

25x = 25

3) Solve the system using elimination.

Step 5: Check your solution.

(1, -1)

3(1) + 4(-1) = -1

4(1) - 3(-1) = 7

3x + 4y = -1

4x – 3y = 7

What is the best number to multiply the top equation by to eliminate the x’s?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32

3x + y = 46x + 4y = 6

1. -4

2. -2

3. 2

4. 4

Solve using elimination.

2x – 3y = 1x + 2y = -3

1. (2, 1)

2. (1, -2)

3. (5, 3)

4. (-1, -1)

Find two numbers whose sum is 18 and whose difference 22.

1. 14 and 4

2. 20 and -2

3. 24 and -6

4. 30 and 8