Solve X and Y in System: 1/2x + 7/2y = 10, -3x + 7y = 12

Solve the above set of equations and choose the correct answer.

$\begin{cases} \frac{1}{2}x+\frac{7}{2}y=10 \\ -3x+7y=12 \end{cases}$

To solve this problem, we'll follow these steps:

• Step 1: Multiply the first equation to make the coefficients of $y$ equal to easily eliminate $y$.
• Step 2: Subtract one equation from the other to eliminate $y$ and solve for $x$.
• Step 3: Substitute the value of $x$ back into one of the original equations to solve for $y$.

Now, let's work through each step:
Step 1: Multiply the first equation $\frac{1}{2}x + \frac{7}{2}y = 10$ by 2 to eliminate fractions:
$x + 7y = 20$

Step 2: Use the second equation as is: $-3x + 7y = 12$. Subtract the equation $x + 7y = 20$ from $-3x + 7y = 12$ to eliminate $y$:
$(-3x + 7y) - (x + 7y) = 12 - 20$
$-4x = -8$
Solve for $x$:
$x = 2$

Step 3: Substitute $x = 2$ back into the equation $x + 7y = 20$:
$2 + 7y = 20$
Subtract 2 from both sides:
$7y = 18$
Divide both sides by 7:
$y = \frac{18}{7} \approx 2.57$

Therefore, the solution that satisfies both equations is $(x, y) = (2, 2.57)$.

The correct choice is $\boxed{x=2, y=2.57}$.