The problem asks to identify which of the given equations (A, B, C, or D) has exactly one solution in common with the equation $y = 6x - 2$.

1. Problem Description

The problem asks to identify which of the given equations (A, B, C, or D) has exactly one solution in common with the equation $y = 6x - 2$.

2. Solution Steps

We are looking for a system of equations (the given equation and one of the choices) that has only one solution. This means the lines are neither parallel (no solution) nor the same line (infinite solutions).

A. $18x - 3y = 6$. Solve for $y$:

$-3y = -18x + 6$

$y = 6x - 2$

This is the same line as the original equation. Thus, they have infinitely many solutions.

B. $\frac{1}{2}y = 3x - 2$. Solve for $y$:

$y = 6x - 4$

This line has the same slope (6) but a different y-intercept (-4) as the original equation $y = 6x - 2$. Therefore, they are parallel lines and have no solution.

C. $2y = 4x - 12$. Solve for $y$:

$y = 2x - 6$

This line has a different slope (2) than the original equation $y = 6x - 2$ (slope is 6). Thus, the lines are not parallel or the same line and will intersect at one point, which means they have exactly one solution.

D. $18x - 12 = 3y$. Solve for $y$:

$3y = 18x - 12$

$y = 6x - 4$

This line has the same slope (6) but a different y-intercept (-4) as the original equation $y = 6x - 2$. Therefore, they are parallel lines and have no solution.

3. Final Answer

C. $2y = 4x - 12$