Score: 50/90 Answered: \( 7 / 9 \) Question 8 Solve the equation \( \frac{x}{4 x-16}-4=\frac{1}{x-4} \) Does the equation have a solution?

To solve the equation \( \frac{x}{4x-16}-4=\frac{1}{x-4} \), let's first simplify it. Notice that \( 4x - 16 \) can be factored to \( 4(x - 4) \). So we rewrite the left-hand side: \[ \frac{x}{4(x-4)} - 4 = \frac{1}{x-4} \] Now, let’s find a common denominator for the left side. The common denominator will be \( 4(x-4) \): \[ \frac{x - 4(4)(x-4)}{4(x-4)} = \frac{1}{x-4} \] Cross-multiplying gives: \[ x - 16(x - 4) = 4 \] Simplifying: \[ x - 16x + 64 = 4 \] Combine like terms: \[ -15x + 64 = 4 \] Now, isolate \( x \): \[ -15x = 4 - 64 \\ -15x = -60 \\ x = 4 \] However, substitute \( x = 4 \) into the original equation. This value leads to division by zero since \( x - 4 = 0 \). Thus, the equation has no solution as \( x = 4 \) is not in the domain of the original expressions.