Algebra 1 Solving equations with variables on both sides. Includes contradiction and identity.

1. VARIABLES ON BOTH
SIDES OF
EQUATIONS
Section 2.4

2. Gather variables together on one side
3𝑥 − 3 = 2𝑥 + 9
−2𝑥−2𝑥
𝑥 − 3 = 9
+3 +3
𝑥 = 12

3. Simplify each side of the equation separately
2 𝑥 + 6 − 𝑥 = 9𝑥 + 11 − 5
2𝑥 + 12 − 𝑥 9𝑥 + 6=
𝑥 + 12 = 9𝑥 + 6
−𝑥 − 𝑥
12 = 8𝑥 + 6
−6 − 6
6 = 8𝑥
÷ 8 ÷ 8
6
8
=
3
4
= 𝑥

4. Do all problems have one correct solution?
How do you respond to 3 = 7 ? No
Contradiction  a false statement
No Real Solution
∅

5. Do all problems have one correct solution?
How do you respond to 4 = 4 ? Yes
Identity  always true statement
All Real Numbers
ℝ

6. Solve the equation
2𝑥 + 3 = 2𝑥 + 7
−2𝑥 − 2𝑥
3 = 7
BUT 3 ≠ 7
So this is a contradiction
No Real Solution ∅

7. Solve the equation
−2 𝑥 + 3 = −2𝑥 − 6
−2𝑥 − 6 = −2𝑥 − 6
We have exactly the same thing on both sides!!!
This is an identity
All Real Numbers
ℝ

8. Solve the equation
−2 𝑥 + 1 = 2(𝑥 − 1)
−2𝑥 − 2 = 2𝑥 − 2
+2𝑥 + 2𝑥
−2 = 4𝑥 − 2
+2 + 2
0 = 4𝑥
÷ 4 ÷ 4
0
4
= 0 = 𝑥
-2+2=0
It is necessary to write down this 0!
You cannot have a nothing blank on either side of =

9. Algebra 1i Homework
Page 100-101 #1-21 odds, 38, 39