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
$y - x = -28$
Answer for screen readers
The value of $y - x$ is $-28$.
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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