Question

Simplify each given equation and show your work. Tell whether it has one solution, an infinite number of solutions, or no solutions, and identify each equation as an identity, a contradiction, or neither. Explain your answers.

(a)6x + 2(2x – 3) = 24 + 9x
(b)25 – 4x = 3(5 – x) + 10 – x
(c)4(x + 2) = 2x + 7 + 2(x – 10)

Answer 1

a) The answer is one solution, neither

6x + 2(2x - 3) = 24 + 9x
6x + 2 * 2x + 2 * (-3) = 24 + 9x
6x + 4x - 6 = 24 + 9x
10x - 6 = 24 + 9x
10x - 9x = 24 + 6
x = 30

b) The answer is infinity solutions, identity
25 – 4x = 3(5 – x) + 10 – x
25 - 4x = 3 * 5 - 3 * x + 10 - x
25 - 4x = 15 - 3x + 10 - x
25 - 4x = 25 - 4x
4x - 4x = 25 - 25
0x = 0

x can be any number, so an infinite number of solution

c) The answer is no solution, contradiction

4(x + 2) = 2x + 7 + 2(x – 10)
4 * x + 4 * 2 = 2x + 7 + 2 * x - 2 * 10
4x + 8 = 2x + 7 + 2x - 20
4x + 8 = 4x - 13
4x - 4x = - 8 - 13
0 = -13

this is contradiction