To solve the system of equations given by
- \(3x + 3y = 6\)
- \(9x - 5y = -24\)
we can use the substitution or elimination method. Let's simplify the first equation:
- Dividing the entire first equation by 3 gives: \[ x + y = 2 \] Hence, we can express \(y\) in terms of \(x\): \[ y = 2 - x \]
Now we can substitute this expression for \(y\) into the second equation:
- Substitute \(y = 2 - x\) into \(9x - 5y = -24\): \[ 9x - 5(2 - x) = -24 \] Distributing \(-5\) gives: \[ 9x - 10 + 5x = -24 \] Combining like terms results in: \[ 14x - 10 = -24 \] Adding 10 to both sides: \[ 14x = -14 \] Dividing both sides by 14 gives: \[ x = -1 \]
Now that we have \(x\), we can find \(y\) using \(y = 2 - x\): \[ y = 2 - (-1) = 2 + 1 = 3 \]
So the solution to the system of equations is: \[ (x, y) = (-1, 3) \]
To ensure our solution is correct, let's plug \(x\) and \(y\) back into the original equations:
For the first equation: \[ 3(-1) + 3(3) = -3 + 9 = 6 \quad \text{(correct)} \]
For the second equation: \[ 9(-1) - 5(3) = -9 - 15 = -24 \quad \text{(correct)} \]
Thus, the solution to the system of equations is: \[ \boxed{(-1, 3)} \]