Solve the Linear System: -x + y = 14 and 5x + 2y = 7

To solve this system of equations, we'll employ the elimination method.

The given system is:

• $-x + y = 14$

• $5x + 2y = 7$

To eliminate $y$, we can multiply the first equation by $2$ to match the coefficient of $y$ in the second equation:

$2(-x + y) = 2 \times 14$

Resulting in:

$-2x + 2y = 28$

Now, we have:

• $-2x + 2y = 28$

• $5x + 2y = 7$

Subtract the second equation from the first:

$\begin{aligned} (-2x + 2y) - (5x + 2y) &= 28 - 7 \\ -2x + 2y - 5x - 2y &= 21 \\ -7x &= 21 \end{aligned}$

Solving for $x$:

$x = \frac{21}{-7} = -3$

Next, substitute $x = -3$ back into the first equation:

$-(-3) + y = 14$

$3 + y = 14$

Solving for $y$:

$y = 14 - 3 = 11$

Therefore, the solution to the system of equations is $x = -3$ and $y = 11$.

The correct answer choice is:

$x=-3,y=11$