Answer
a. Yes, x=2 is a solution to the given equation.
b. No, x=4 is not a solution to the given equation.
Work Step by Step
Simplify the original equation:
1. $\frac{1}{x}-\frac{1}{x-4}=1$
2. Use the formula $\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$ to simplify the equation: $\frac{x-4-x}{x^{2}-4x} = 1$
3. Simplify: $\frac{-4}{x^{2}-4x} = 1$
4. Multiply ${x^{2}-4x}$ by 1 to simplify: $-4={x^{2}-4x}$
5. Add -4 to both sides: $0=x^{2}-4x+4$
6. Factor: $0=(x-2)(x-2)$
a. Plug in x=2 to see if it makes the equation true: $0=(2-2)(2-2)$ $0=0$ The equation is true so x=2 is a solution of the given equation.
b. Plug in x=4 to see if it makes the equation true: $0=(4-2)(4-2)$ $0=4$ is not true, so 4 is not a solution of the given equation.
