1. 2-4, 2-5 Warm Up 1 1 3

2. Algebra 1 Glencoe McGraw-Hill JoAnn Evans
Math 8H
2-4 and 2-5
Solving Equations
With Variables on Both Sides &
Special Cases
Algebra Glencoe McGraw-Hill JoAnn Evans

3. When solving equations with variables on both sides, there are two schools of thought:
Collect variables on the left:
Collect variables on the side with the greater variable coefficient:
x = 2x – 6
x + 4 = 2x – 6
-2x x
-x x
-x = -6
4 = x - 6
-x = -10
10 = x
x = 10

4. 7x = -2x + 55
Does the solution of 4 make a true statement when substituted in the original equation?
+2x x
9x = 55
9x = 36
9x = 36
x = 4

5. 4(1 – x) + 3x = -2(x + 1)
4 - 4x + 3x = -2x - 2
4 - x = -2x - 2
+2x x
4 + x = -2
x = -6

6. ¼(12x + 16) = (x – 2)
3x = x + 6
3x = x
+3x x
6x = 16
6x = 12
6x = 12
x = 2

7. 7(2 – x) = 5x
Leave your answer in the form of an improper fraction. In algebra, that’s the preferred form instead of a mixed number.
x = 5x
+ 7x x
14 = 12x 1
14 = 12x

8. Do you notice anything unusual about this equation?
A special type of equation
Do you notice anything unusual about this equation?
3(4 – x) = -2x x
12 - 3x = -3x + 12
-3x + 12 = -3x + 12
Both sides of the equation are identical. When this happens it’s called an identity. Keep on solving the equation.

9. Write: All Real Numbers
-3x + 12 = -3x + 12
In an “identity” all variable terms will cancel out, leaving you with a true statement such as this one.
+3x x
12 = 12
An identity will have MANY SOLUTIONS. When the two sides of an equation are identical, any number you substituted as a solution would make a true statement. All real numbers are solutions of an identity.
Write: All Real Numbers

10. Another special type of equation
8x – [4 – (-2)] = 8x
Untrue statement!
8x – [4 + 2] = 8x
8x – 6 = 8x
-8x x
-6 = 0
When all variable terms cancel out and what’s left is an untrue statement of equality, there is no solution to the equation. Write: No Solution

11. Solve the equation, if possible
Solve the equation, if possible. Determine whether it has one solution, no solution, or all real numbers are solutions.
The two sides of the equation are identical. It is an identity. All real numbers are solutions of this equation.

12. An untrue statement is left, so there is
no solution to this equation. No value of the variable will make the equation true.
This equation has one solution.
x = -10

13. One solution!

14. Identity
No solution!
All real numbers!

15. Consecutive Integer Problems
Consecutive integer means one right after another:
Ex: 3, 4, Odds: 3, 5, 7 Evens: 4, 6, 8
Your first let statement is ALWAYS “Let x = first number”
Your second let statement is “Let x + 1 = second number”
OR “Let x + 2 = second number”
Your third let statement is “Let x + 2 = third number”
OR “Let x + 4 = third number”

16. Consecutive Integer Problems
Find three consecutive integers with a sum of 87.
x = first number x + 1 = second number
x + 2 = third number
Verbal: first # + second # + third # = Total
Equation: x + x x + 2 = 87
3x + 3 = 87
3x = 84
x = 28
The numbers are 28, 29, 30

17. Consecutive Integer Problems
Find three consecutive EVEN integers with a sum of 126.
x = first number x + 2 = second number
x + 4 = third number
Verbal: first # + second # + third # = Total
Equation: x + x x + 4 = 126
3x + 6 = 126
3x = 120
x = 40
The numbers are 40, 42, 44