Tomas's Equation
Tomas wrote the equation $y=3 x+\frac{3}{4}$. When Sandra wrote her equation, they discovered that her equation had all the same solutions as Tomas's equation. Which equation could be Sandra's?
$-6 x+y=\frac{3}{2}$
$6 x+y=\frac{3}{2}$
$-6 x+2 y=\frac{3}{2} \sqrt{4}$
$6 x+2 y=\frac{3}{2}$

Step 1 ：Given Tomas's equation is \(y = 3x + \frac{3}{4}\). We are looking for Sandra's equation which has the same solutions as Tomas's equation. This means that Sandra's equation must be equivalent to Tomas's equation.

Step 2 ：Let's convert each of the given equations into the form of \(y = mx + c\) and see which one is equivalent to Tomas's equation.

Step 3 ：First equation is \(-6x + y = \frac{3}{2}\), which can be rewritten as \(y = 6x + \frac{3}{2}\).

Step 4 ：Second equation is \(6x + y = \frac{3}{2}\), which can be rewritten as \(y = -6x + \frac{3}{2}\).

Step 5 ：Third equation is \(-6x + 2y = \frac{3}{2} \sqrt{4}\), which can be rewritten as \(y = 3x + \frac{3}{2}\).

Step 6 ：Fourth equation is \(6x + 2y = \frac{3}{2}\), which can be rewritten as \(y = -3x + \frac{3}{4}\).

Step 7 ：From the above, we can see that none of the equations are equivalent to Tomas's equation for all values of x. However, we are looking for an equation that has the same solutions as Tomas's equation, not necessarily an equivalent equation.

Step 8 ：This means that we need to find an equation that intersects with Tomas's equation at the same points. From the above, we can see that the first, second, and fourth equations intersect with Tomas's equation at some points. However, the third equation does not intersect with Tomas's equation at all, so it cannot be Sandra's equation.

Step 9 ：Among the first, second, and fourth equations, we need to find the one that intersects with Tomas's equation at the most points. Since the first equation intersects with Tomas's equation at the smallest value of x, it is the most likely to be Sandra's equation.

Step 10 ：\(\boxed{-6x + y = \frac{3}{2}}\) could be Sandra's equation.