1. TYPES OF SOLUTIONS OF LINEAR EQUATIONS
Algebraic Form
Number of Solutions
Description
a = b
None
There are no values of the variable for which the equation is true.
x = a
One
The equation is true for exactly one value of the variable.
a = a
Infinitely many
The equation is true for all values of the variable.

2. 1. 2x – 4 = -x - 1 + x + x 3x – 4 = - 1 + 4 + 4 3x = 3 3 = 3 x = 1
Tell whether each equation has one solution, infinitely many solutions, or no solutions. Justify your answer.
x – 4 = -x - 1
+ x x
Add 1x to each side
Simplify
Add 4 to each side
Divide both sides by 3
3x – 4 = - 1
3x = 3
= 3
x = 1
The result is an equation of the form x = a. This equation is true for exactly one value. So, the equation has one solution.

3. Tell whether each equation has one solution, infinitely many solutions, or no solutions. Justify your answer.
x – 4 = 2(x – 2)
Distributive Property
Subtract 2x from both sides
Simplify
2x – 4 = 2x – 4
- 2x x
– 4 = - 4
The result is an equation of the form a = a. This equation is true for all values of x. So, the equation has infinitely many solutions.

4. Tell whether each equation has one solution, infinitely many solutions, or no solutions. Justify your answer.
x – 4 = 2(x + 1)
Distributive Property
Subtract 2x from both sides
Simplify
2x – 4 = 2x + 2
- 2x x
– 4 = 2
The result is an equation of the form a = b. There are no values of x for which the equation is true. So, the equation no solution.

5. a. 5x + 8 = 5(x + 3) b. 9x = 8 + 5x - 5x - 5x 5x + 8 = 5x + 15
Tell whether each equation has one solution, infinitely many solutions, or no solutions. Justify your answer.
a x + 8 = 5(x + 3)
b x = 8 + 5x
- 5x x
5x + 8 = 5x + 15
- 5x x
4x = 8
8 = 15
x = 2
no solution
form a = b
one solution
form x = a

6. c. 6x + 12 = 6(x + 2) d. 7x - 11 = 11 - 7x + 7x + 7x 6x + 12 = 6x + 12
Tell whether each equation has one solution, infinitely many solutions, or no solutions. Justify your answer.
c x + 12 = 6(x + 2)
d x - 11 = x
+ 7x x
6x + 12 = 6x + 12
- 6x x
14x - 11 = 11
12 = 12
14x = 22
x = = 1 4 7
infinitely many solutions
form a = a
one solution
form x = a