To solve the system of equations
- \( 6x - 3y = -6 \)
- \( 9x - 3y = -18 \)
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:
- \( y = 2 + 2x \)
- \( y = 6 + 3x \)
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:
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For \(6x - 3y = -6\): \[ 6(-4) - 3(-6) = -24 + 18 = -6 \quad \text{(True)} \]
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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: \[ \boxed{( -4, -6 )} \]