Questions: Solve the following equation. (3x-2)/(x-3)=(7)/(x-3) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is ∅.

$Solve the following equation. (3x-2)/(x-3)=(7)/(x-3) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is ∅.$

Transcript text: Solve the following equation. \[ \frac{3 x-2}{x-3}=\frac{7}{x-3} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ \}. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is $\varnothing$.

Solution

Solution Steps

To solve the given equation, first observe that the denominators on both sides are the same. This allows us to equate the numerators directly. Solve the resulting linear equation for $ x $. Check if the solution makes the original denominators zero, as such values would be excluded from the solution set.

Step 1: Set Up the Equation

We start with the equation

\[ \frac{3x - 2}{x - 3} = \frac{7}{x - 3} \]

Since the denominators are the same, we can equate the numerators:

\[ 3x - 2 = 7 \]

Step 2: Solve for $ x $

Next, we solve the equation $ 3x - 2 = 7 $:

\[ 3x = 7 + 2 \] \[ 3x = 9 \] \[ x = \frac{9}{3} = 3 \]

Step 3: Check for Validity

We need to check if the solution $ x = 3 $ makes the original denominator zero. The denominator is $ x - 3 $, and substituting $ x = 3 $ gives:

\[ 3 - 3 = 0 \]

Since the denominator cannot be zero, $ x = 3 $ is not a valid solution.