Question

What is the value of y in the solution to the system of equations?

One-thirdx + One-fourthy = 1

2x – 3y = –30

–8
–3
3
8

Answer 1

To find the value of \(y\) in the system of equations:

1. \(\frac{1}{3}x + \frac{1}{4}y = 1\)
2. \(2x - 3y = -30\)

We can first eliminate the fractions in the first equation. Multiply the entire first equation by 12 (the least common multiple of 3 and 4):

\[
12 \left( \frac{1}{3}x \right) + 12 \left( \frac{1}{4}y \right) = 12 \cdot 1
\]

This simplifies to:

\[
4x + 3y = 12 \quad \text{(Equation 1)}
\]

Now we have a new system of equations:

1. \(4x + 3y = 12\)
2. \(2x - 3y = -30\)

Next, we can add the two equations together to eliminate \(y\):

\[
(4x + 3y) + (2x - 3y) = 12 - 30
\]

This simplifies to:

\[
6x = -18
\]

Now, solving for \(x\):

\[
x = -3
\]

Now that we have the value of \(x\), we can substitute it back into one of the equations to find \(y\). We'll use Equation 1:

\[
4(-3) + 3y = 12
\]

This simplifies to:

\[
-12 + 3y = 12
\]

Adding 12 to both sides gives:

\[
3y = 24
\]

Now, dividing by 3:

\[
y = 8
\]

Thus, the value of \(y\) is \(\boxed{8}\).