Question

Solve the system of equations.

\[\begin{aligned}
&-2x-7y = 30
\\\\
&7x+4y=18
\end{aligned}\]
\[x=\]
\[y=\]

Answer 1

To solve the system of equations:

\[
\begin{aligned}
-2x - 7y &= 30 \quad \text{(1)} \\
7x + 4y &= 18 \quad \text{(2)}
\end{aligned}
\]

we will use the substitution or elimination method. Here, I will use the elimination method.

First, let's manipulate the equations to eliminate \(y\). We can multiply equation (1) by 4 and equation (2) by 7 to align the coefficients of \(y\):

\[
\begin{aligned}
4(-2x - 7y) &= 4(30) \\
-8x - 28y &= 120 \quad \text{(3)}
\end{aligned}
\]

And for equation (2):

\[
\begin{aligned}
7(7x + 4y) &= 7(18) \\
49x + 28y &= 126 \quad \text{(4)}
\end{aligned}
\]

Now we have a new system of equations:

\[
\begin{aligned}
-8x - 28y &= 120 \quad \text{(3)} \\
49x + 28y &= 126 \quad \text{(4)}
\end{aligned}
\]

Next, we can add equations (3) and (4) together to eliminate \(y\):

\[
(-8x - 28y) + (49x + 28y) = 120 + 126
\]

This simplifies to:

\[
41x = 246
\]

Now, divide both sides by 41:

\[
x = \frac{246}{41} = 6
\]

Now that we have \(x\), we can substitute it back into one of the original equations to find \(y\). We can use equation (2):

\[
7x + 4y = 18
\]

Substituting \(x = 6\):

\[
7(6) + 4y = 18
\]

This simplifies to:

\[
42 + 4y = 18
\]

Subtract 42 from both sides:

\[
4y = 18 - 42
\]
\[
4y = -24
\]

Now divide by 4:

\[
y = \frac{-24}{4} = -6
\]

The solution of the system of equations is:

\[
x = 6, \quad y = -6
\]

Thus, the final answer is:

\[
\boxed{x = 6, \quad y = -6}
\]