An equation is shown below. \[ 3(x-2)+7 x=\frac{1}{2}(6 x-2) \] How many solutions, if any. does the equation have?

To find the number of solutions for the equation \( 3(x-2)+7 x=\frac{1}{2}(6 x-2) \), let's start by simplifying it step by step. 1. Distribute on the left side: \[ 3x - 6 + 7x = \frac{1}{2}(6x - 2) \] Combine like terms: \[ 10x - 6 = \frac{1}{2}(6x - 2) \] 2. Now simplify the right side: \[ \frac{1}{2}(6x - 2) = 3x - 1 \] 3. Now the equation is: \[ 10x - 6 = 3x - 1 \] 4. Move all terms involving \( x \) to one side and constant terms to the other side: \[ 10x - 3x = -1 + 6 \] \[ 7x = 5 \] 5. Solve for \( x \): \[ x = \frac{5}{7} \] Since we found a unique solution \( x = \frac{5}{7} \), the equation has exactly one solution!