Transform each of the following equations to determine whether each has one solution, infinitely many solutions or no solution. Use the result of your transformation to state the number of solutions.

1. 2x + 4 = 3 (x - 2) + 1

2. 4 (x + 3) = 3x + 17

3. 3x + 4 + 2x = 5 (x - 2) + 7

4. 2 (1 + 5x) = 5 (2x - 1)

5. - 18 + 15x = 3 (4x - 6) + 3x

1.

we distribute then move placeholders to get

2x+4=4x-6+1

9=x

one solution

2. distribute

4x+12=3x+17

x=5

one solution

3. distribute and combine like terms

5x+4=5x-10+7

4=-3

false

no solution

4. distribute

2+10x=10x-5

2=-5

false

no solution

5. distribute

-18+15x=12x-18+3x

x=x

true

infinite solutions

1. 2x + 4 = 3 (x - 2) + 1

2x + 4=3x-6+1

2x-3x=-5-4

-x = - 9

x = 9 one solution

2. 4 (x + 3) = 3x + 17

4x + 12=3x+17

4x-3x=17-12

x = 5 one solution

3. 3x + 4 + 2x = 5 (x - 2) + 7

5x+4=5x-10+7

5x-5x=-3-4

0x=-7 no solution

4. 2 (1 + 5x) = 5 (2x - 1)

2+10x=10x-5

10x-10x=-5-2

0x = - 7 no solution

5. - 18 + 15x = 3 (4x - 6) + 3x

-18+15x=12x-18+3x

15x-15x = - 18+18

0 = 0 infinite solutions