1. Solving Equations with Variables on Both Sides
Essential Question?
How can you solve equations with variables on both sides and identify special cases?
8.EE.7a

2. Common Core Standard:
8.EE. ─ Analyze and solve linear equations and pairs of simultaneous linear equations..
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

3. Objective - To solve equations with variables on both sides.

4. Not all equations can be solved.
SPECIAL CASES
Not all equations can be solved.
Most of the time, you will find the equation has ONE SOLUTION. This is the most common situation.
3x - 5 = 2x + 12
-2x x
x - 5 = 12
x = 17
One Solution

5. SPECIAL CASES
Sometimes both sides of the equation are IDENTICAL to each other.
i.e. 2x = 2x
This is called an IDENTITY.
Think about how many numbers you can substitute for x that will make the equation true.
An IDENTITY has INFINITE solutions.

6. When there are an infinite number of solutions, we say the solution is
INFINITE SOLUTIONS
When there are an infinite number of solutions, we say the solution is
x = all real numbers
Steps
Reason
Given
3x + 8 = 2(x + 4) + x
3x + 8 = 2x x
3x + 8 = 3x + 8
-3x x
8 = 8
True !
Identity
Infinite solutions
x = all real numbers

7. SPECIAL CASES
Sometimes the variables on both sides of the equation cancel each other out to make an equation that is false.
i.e. 2 = 7
This is NOT TRUE.
Nothing you can do will make the equation true.
A NOT TRUE equation has NO SOLUTION.

8. When there are no solutions, we say the answer is
Steps
Reason
Given
3x + 2 = 2(x - 1) + x
3x + 2 = 2x x
3x + 2 = 3x - 2
-3x x
2 = -2
False !
Not Identical
Not True
No Solution

9. Solve: 4(y - 2) + 6y = 7(y - 8) - 3(10 - y)
-8 = -86
False Statement
No Solution

10. Solve: 3(4 + k) - 2(3k + 4) = 5(k - 3) - (8k - 19)
4 = 4
True Statement
Infinitely Many Solutions
k = all real numbers

11. Use a variable equation to solve:
APPLICATION
Use a variable equation to solve:
Find a number that is 20 more than 4 times its opposite.
Let x = the number
-x = the opposite
x = 4(-x) + 20
x = -4x + 20
+4x +4x
5x = 20
x = 4