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Solving Systems of Equations
by Elimination
Warm Up
4/3/17
Solve each system by graphing.
1. 𝑦 =
1
𝑥
2
+ 1.5
𝑦 = −2𝑥 + 4
(1, 2)
2. 3𝑥 + 𝑦 = −5
𝑦 = −5
(0, -5)
Elimination To get rid of.
When 2 variables have the same coefficients, they can be
eliminated by subtraction. (ex. 2x – 2x = 0
When 2 variables have the same inverse coefficients, they can
be eliminated by addition. (ex. 3y + -3y = 0)
Independent Discovery
Randy has some $1 bills and $5 bills in his wallet. He has 15 bills in all. He
counts the money and finds he has $47. How many of each type of bill does
Randy have?
Write a system of equations for this problem.
Step 1: Define the variables
x = number of $1 bills
y = number of $5 bills
Step 2: Write the equations
1st equations represents the total amount of bills:
𝑥 + 𝑦 = 15
2nd equation represents the amount of money:
1𝑥 + 5𝑦 = 47
Solving by Elimination
1. Add the equations by adding the like terms. Describe the
result. Can you solve the resulting equation? Why or why
not?
𝑥 + 𝑦 = 15
𝑥 + 5𝑦 = 47
2. Subtract the equations by subtracting the like terms.
Describe the result. Can you solve the resulting equation?
Why or why not?
Solving by Elimination (add/subtract)
Step 1: Write each equation in Standard Form
Ax + By = C
Step 2: Add or subtract the equations. Remember to
distribute the subtraction symbol to all terms.
Step 3: Substitute the value of the variable you solved for
back into an original equation. Solve for the other
variable.
Step 4: Check by substituting both answers into both
equations.
Solving a System by Adding
Step 1: Eliminate one variable. Since the coefficients of y
are additive inverses, add to eliminate y.
5x – 6y = -32
+ 3x + 6y = 48
8x + 0 = 16
x=2
5x – 6y = -32
3x + 6y = 48
Solution: (2,7)
 Add
 Solve for x (Divide both sides by 8)
Step 2: Substitute 2 for x in either equation and solve for y.
5x – 6y = -32  Write either equation
5(2) – 6y = -32  Substitute 2 for x
10 – 6y = -32  Subtract 10 from both sides
-6y = -42  Simplify
y = 7  Solve for y (Divide both sides by -6)
Solving a System by Subtracting
At the school store, Ricardo bought 4 pencils and 6 erasers and
spent $2.60. Annabelle bought 4 pencils and 10 erasers and spent
$3.80. Solve the system of equations to determine the cost of 1
pencil and the cost of 1 eraser.
Step 1: Write the system of equations. Let p = the cost of each pencil,
and let e = the cost of each eraser.
4p + 6e = 2.60  amount Ricardo spent
4p + 10e = 3.80  amount Annabelle spent
Step 2: Subtract the equations. Since the coefficients of p are the same, subtract
to eliminate p.
4p + 6e = 2.60
- (4p + 10e = 3.80)
The cost of one
eraser is $0.30
0 - 4e = - 1.20  Subtract
e = 0.30  Solve for e (Divide both sides by -4)
Step 2: Substitute 0.30 for e in either equation and solve for p.
4p + 10e = 3.80  Write either equation
4p + 10(0.30) = 3.80  Substitute 0.30 for e
The cost of one
4p + 3 = 3.80  Simplify
pencil
is
$0.20
4p = 0.80  Subtract 3 from each side
p = 0.20  Solve for y (Divide both sides by 4)
Solving a System by Subtracting
Example 2
Franklin and Marianne sell gourmet cakes for a fundraiser.
Franklin sells 2 large cakes and 6 small cakes for $190. Marianne
sells 2 large cakes and 3 small cakes for $130. Find the cost of
each small cake.
Step 1: Write the system of equations. Let l = the cost of each large
cake, and let s = the cost of each small cake.
2l + 6s = 190  amount Franklin raised
2l + 3s = 130  amount Marianne raised
Step 2: Subtract the equations. Since the coefficients of l are the same, subtract
to eliminate l.
2l + 6s = 190
- (2l + 3s = 130)
0 + 3s = 60
 Subtract
s = 20  Solve for e (Divide both sides by 3)
The cost of one small cake is $20
Examples 1 and 2 show that in order to eliminate a variable, the
coefficients of the variable must be the same or the additive inverse.
Sometimes you have to multiply each side of one or both of the equations in a
system by a nonzero number before you can eliminate a variable. (We will look
at those examples tomorrow)
CW/HW – Solve each system using elimination. Check your solution.
1. 𝑥 + 𝑦 = 9
𝑥−𝑦 =1
3. −𝑥 + 𝑦 = −2
3𝑥 − 𝑦 = 4
2. 3𝑥 + 2𝑦 = 2
𝑥 − 2𝑦 = 6
4. 3𝑥 + 𝑦 = 9
3𝑥 + 3𝑦 = 21
Solving Systems of Equations using
Elimination
Day 2
Warm Up
4/4/17
Solve each system by graphing.
1. 2𝑥 − 𝑦 = 1
𝑥 =𝑦+1
(0, -1)
2. 8𝑥 − 4𝑦 + 8 = 0
6x + 3𝑦 + 6 = 0
(-1, 0)
Solving a Linear Equation by Multiplying First:
Step 1: Write each equation in Standard Form (Ax + By = C)
Step 2: Multiply/ divide one or both equations so one variable
has identical or inverse coefficients
Step 3: Add or subtract the equations. Remember to
distribute the subtraction symbol to all terms.
Step 4: Substitute the value of the variable you solved for back
into an original equation. Solve for the other variable.
Step 5: Check by substituting both answers into both
equations.
Solving a System by Multiplying
Step 1: To eliminate x, multiply each term of the
second equation by 2. Then add.
10x – 7y = 2
2(-5x + 3y = -3) = 2(-3)


10x – 7y = 2
-5x + 3y = -3
10x – 7y = 2
+ (-10x + 6y = -6)
0 – 1y = -4  Add
y = 4  Solve for y (Divide both sides by -1)
Step 2: Substitute 4 for y in either original equation and solve for x.
10x – 7y = 2  Write either equation
10x – 7(4) = 2  Substitute 4 for y
Solution: (3,4)
10x – 28 = 2 Simplify
10x = 30 Add 28 to both sides
x = 3  Solve for x (Divide both sides by 10)
Solve each system of equations by elimination. Show your work.
1.
2x + 6y = -22
4x – 3y = -14
(2, ½)
2.
x – 4y = 2
3x + 5y = 40
(10, 2)
3.
4x – 2y = 7
3x + 6y = 9
(2, ½)
1. Which of the following systems would be most efficiently solved using the
elimination method?
System A:
2x + 2y = 6
-6x – 2y = 6
System B:
y = 2 – 3x
4x – 2y = -2
System A – the coefficients of y are additive inverses of each other.
2. Explain how you would solve the following system using the elimination
method.
2x – 5y = -6
2x – 7y = 14
Subtract the second equation from the first to eliminate x.
Solve each system of equations by elimination. Show your work.
1.
2x + y = 12
x–y=3
(5, 2)
2.
How do you know when to add the equations to eliminate a
variable?
When the coefficients of the
same variable are positive and
negative
Exit Ticket
On a separate sheet of paper to turn in.
1. Solve the system by elimination and check your
solution.
5𝑥 − 6𝑦 = 1
2𝑥 + 2𝑦 = 18
2. Would you add or subtract the equations to solve the
following system? Explain your reasoning.
5𝑥 + 3𝑦 = 4
5𝑥 − 3𝑦 = −16