Name
____________________________________
11.3 Solving Systems of Equations with the Elimination Method
Per ________
We can also solve systems algebraically another way besides the Substitution method. It is called the Elimination
method. It is the process of adding/subtracting two equations together resulting in an equation with one variable.
Once you have solved for the variable you can use it to find the other.
Elimination Steps:
1. Place both equations in Standard Form, Ax + By = C.
2. Determine which variable to eliminate with Addition or Subtraction.
3. Solve for the variable left.
4. Go back and use the found variable in step 3 to find second variable.
5. Check the solution in both equations of the system.
Let’s try it out, solve the following system of equations: -4x - 2y = -12
4x + 8y = -24
Step 1. Place both equations in Standard Form
Step 2: Determine which variable to eliminate with
Addition or Subtraction..
Lucky us! Both equations are in standard form
Looks like x is set up to be eliminated if I add the two
equations together (straight down).
Step 3: Add or Subtract and solve for the variable
-4x - 2y = -12 » 6y = -36 » y = -6
4x + 8y = -24
Step 4: Solve for the variable left (x) by plugging it
into one of the equations.
-4x – 2(-6) = -12 » -4x +12 = -12 » -4x = -24 » y = 6
So Solution is (6, -6)
Step 5: Check the Solution in both equations.
-4(6) – 2(-6) = -12 » -24 + 12 = -12 CHECK
4(6) + 8(-6) = -24
» 24 – 48 = -24 CHECK
We can also determine algebraically if a system of Equations has one solution, no solutions or infinitely many solutions.
ONE SOLUTION (intersecting lines)
Variables will have an answer
ex. x = 2; y = 3 » (2,3)
NO SOLUTION (parallel lines)
Variables will drop out and a false statement is left ex. 4 = 2
INFINITELY MANY SOLUTIONS
(coincidental lines: same line)
Variables will drop out and a true statement is left ex. 4 = 4
Homework
Solve the following systems of equations algebraically using the Elimination Method. They will have one, none or
infinitely many solutions.
1.
4𝑥 + 𝑦 = 30
−4𝑥 + 3𝑦 = −6
2.
𝑥 + 3𝑦 = −18
−𝑥 − 2𝑦 = 12
3.
4𝑥 + 𝑦 = 12
4𝑥 + 3𝑦 = +4
4.
−3𝑥 + 𝑦 = 11
−3𝑥 − 4𝑦 = 31
5.
4𝑥 − 𝑦 = 4
4𝑥 − 𝑦 = 12
6.
2𝑥 − 3𝑦 = −11
6𝑥 − 9𝑦 = −33
7.
𝑥 + 2𝑦 = −5
−𝑥 − 3𝑦 = 11
8.
3𝑥 + 3𝑦 = −21
3𝑥 − 2𝑦 = 14
9.
𝑥 − 5𝑦 = 26
𝑥 + 5𝑦 = −24
10.
−4𝑥 + 2𝑦 = −46
3𝑥 − 2𝑦 = 37
12.
−4𝑥 + 4𝑦 = 36
−𝑥 + 𝑦 = 9
11.
4𝑥 + 2𝑦 = −2
4𝑥 − 𝑦 = 13
13.
– 𝑥 − 7𝑦 = 14
−4𝑥 − 14𝑦 = 28
14.
−4𝑥 − 15𝑦 = −17
−𝑥 + 5𝑦 = −13
15.
– 7𝑥 + 𝑦 = −19
−2𝑥 + 3𝑦 = −19
16.
−4𝑥 + 9𝑦 = 9
𝑥 − 3𝑦 = −6
Solve the systems of equations
by graphing
3
4
17. 𝑦 = − 𝑥 − 1
3
4
18. 𝑥 = −4
𝑥 − 2𝑦 = −6
𝑦 =− 𝑥−3
20. Solve using Substitution:
19. Fill in the blank.
a. Parallel lines have _______
solutions.
a. 5𝑥 = 2 − 𝑦
𝑥 = 4+𝑦
b. Lines with different slopes
have ________ solution.
c. Coincidental Lines have
__________ solutions.
Key:
1. (6,6)
3. (4,-4)
5. No Sol.
7. (7, -6)
9. (1, -5)
11. (2,-5)
13. (0, -2)
15. (2, -5)
17. No Solution
b. 4 = 𝑥 − 2𝑦
𝑦 = 2𝑥 + 1