Question

Solve using substitution.

–6x − 9y = 3
2x + 2y = –6

Answer 1

To solve the system of equations using substitution, we'll start with the two given equations:

1. \(-6x - 9y = 3\)
2. \(2x + 2y = -6\)

First, we can simplify the second equation by dividing everything by 2:

\[
x + y = -3
\]

Now we can solve for \(y\) in terms of \(x\):

\[
y = -3 - x
\]

Next, we'll substitute this expression for \(y\) into the first equation:

\(-6x - 9y = 3\)

Substituting \(y\):

\[
-6x - 9(-3 - x) = 3
\]

Now we simplify the left side:

\[
-6x + 27 + 9x = 3
\]

Combining like terms gives:

\[
3x + 27 = 3
\]

Next, we isolate \(x\) by subtracting 27 from both sides:

\[
3x = 3 - 27
\]
\[
3x = -24
\]

Now, divide by 3:

\[
x = -8
\]

Now that we have \(x\), we can substitute it back to find \(y\):

\[
y = -3 - (-8)
\]
\[
y = -3 + 8
\]
\[
y = 5
\]

Thus, the solution to the system of equations is:

\[
(x, y) = (-8, 5)
\]

To verify, we can substitute these values back into the original equations:

1. For \(-6(-8) - 9(5) = 3\):

\[
48 - 45 = 3 \quad \text{(True)}
\]

2. For \(2(-8) + 2(5) = -6\):

\[
-16 + 10 = -6 \quad \text{(True)}
\]

Both equations hold true, confirming our solution is correct:

\(\boxed{(-8, 5)}\)