Question

Solve the system of equations.
6x−3y=−6

9x−3y=−18
(1 point)

Answer 1

To solve the system of equations

1) \( 6x - 3y = -6 \)
2) \( 9x - 3y = -18 \)

we can start by observing these equations. Notice that both equations have a common term, \(-3y\).

Let's first rearrange these equations to identify their relationships.

**From the first equation (1):**
\[
6x - 3y = -6
\]
We can express it in terms of \(y\):
\[
-3y = -6 - 6x \implies 3y = 6 + 6x \implies y = 2 + 2x
\]

**From the second equation (2):**
\[
9x - 3y = -18
\]
Similarly, we can express it in terms of \(y\):
\[
-3y = -18 - 9x \implies 3y = 18 + 9x \implies y = 6 + 3x
\]

Now we have:
1) \( y = 2 + 2x \)
2) \( y = 6 + 3x \)

Next, we can set these two expressions for \(y\) equal to each other:
\[
2 + 2x = 6 + 3x
\]

To solve for \(x\), we rearrange the equation:
\[
2 + 2x - 3x = 6
\]
\[
2 - x = 6
\]
\[
-x = 6 - 2 \implies -x = 4 \implies x = -4
\]

Now that we have \(x\), we can substitute \(x = -4\) back into either equation to find \(y\). We'll use the first equation:
\[
y = 2 + 2(-4) = 2 - 8 = -6
\]

Thus, the solution to the system of equations is:
\[
\boxed{( -4, -6 )}
\]

To verify, we can substitute \(x = -4\) and \(y = -6\) into both original equations:

1) For \(6x - 3y = -6\):
\[
6(-4) - 3(-6) = -24 + 18 = -6 \quad \text{(True)}
\]

2) For \(9x - 3y = -18\):
\[
9(-4) - 3(-6) = -36 + 18 = -18 \quad \text{(True)}
\]

Both equations are satisfied, confirming that the solution is correct. The final solution is:
\[
\boxed{( -4, -6 )}
\]