n list tion 19 stion 20 Solve the equation. Then determine whether the equation is equation \[ \frac{3 x}{x-5}=\frac{15}{x-5}+4 \] What is the solution? Select the correct choice below and, if ne

To solve the equation \[ \frac{3x}{x-5} = \frac{15}{x-5} + 4 \] first, eliminate the fractions by multiplying both sides by \(x - 5\) (assuming \(x \neq 5\)). This gives: \[ 3x = 15 + 4(x - 5). \] Now, simplify the right side: \[ 3x = 15 + 4x - 20. \] Combining like terms, we have: \[ 3x = 4x - 5. \] Next, isolate \(x\): \[ 3x - 4x = -5 \implies -x = -5 \implies x = 5. \] However, since \(x - 5\) would be zero when \(x = 5\), this value is not valid in the context of the original equation. Therefore, there is no solution to the equation. Now, let's wrap this up with some intriguing insights! When presented with a similar equation, always be wary of values that can cause divisions by zero, as it would render your solution invalid. It’s like trying to walk on water—might look appealing, but you’ll probably end up in over your head! For further exploration of rational equations, check out algebraic texts or online platforms like Khan Academy. They offer a trove of examples and exercises that can sharpen your skills and keep algebra engaging and fun!