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

Understand the Problem

The question is asking for the value of the expression y - x based on the provided system of equations. To solve this, we will first solve the system to find the values of x and y.

Answer

$y - x = -28$

Steps to Solve

Rearranging the first equation

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

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

Substituting into the second equation

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

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

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: $$ 56 = -16 - 3y $$

Final steps to isolate y

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

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

Finding x

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

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

Calculating y - x

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

The value of $y - x$ is $-28$.

More Information

This solution involves solving a system of linear equations using substitution. It’s a common method in algebra and is useful for finding specific values in equations.

Tips

Miscalculating while substituting values.
Forgetting to change signs when moving terms to the other side of the equation.
Not fully isolating $y$ before calculating $x$.

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