For the system shown, what is the value of y - x?

Steps to Solve

1. Rearranging the first equation

Start with the first equation: $$ x + \frac{3}{4}y = -14 $$

Rearranging it gives: $$ x = -14 - \frac{3}{4}y $$

2. Substituting into the second equation

Now substitute $x$ into the second equation: $$ -4x + 3y = -16 $$

Substituting for $x$: $$ -4(-14 - \frac{3}{4}y) + 3y = -16 $$

3. Solving for y

Distributing the $-4$: $$ 56 + 3y = -16 + 3y $$

Now, combine like terms: $$ 56 + 3y = -16 $$

Subtracting $3y$ from both sides helps isolate terms: $$ 56 = -16 - 3y $$

4. Final steps to isolate y

Add $16$ to both sides: $$ 72 = -3y $$

Dividing both sides by $-3$ gives: $$ y = -24 $$

5. Finding x

Now substitute $y$ back into the rearranged first equation: $$ x = -14 - \frac{3}{4}(-24) $$

Calculate $\frac{3}{4}(-24)$: $$ x = -14 + 18 = 4 $$

6. Calculating y - x

Now that we have both $x$ and $y$, we can calculate $y - x$: $$ y - x = -24 - 4 = -28 $$