To solve the system of equations:
-
\(-5x + 7y = 67\)
-
\(11x + 12y = 17\)
we can use substitution or elimination. Let's use the elimination method.
First, we can multiply the first equation by a suitable coefficient so that the coefficients of \(x\) in both equations can be easily eliminated. Let's multiply the first equation by \(11\) and the second equation by \(5\) to align the \(x\) coefficients.
Multiplying the first equation by \(11\):
\[ -55x + 77y = 737 \]
Multiplying the second equation by \(5\):
\[ 55x + 60y = 85 \]
Now, we can add both equations:
\[ (-55x + 77y) + (55x + 60y) = 737 + 85 \]
The \(x\) terms cancel out:
\[ (77y + 60y) = 822 \]
Combining \(y\) terms gives:
\[ 137y = 822 \]
Divide both sides by \(137\):
\[ y = \frac{822}{137} = 6 \]
Now that we have \(y\), we can substitute it back into one of the original equations to find \(x\). Using the first equation:
\(-5x + 7(6) = 67\)
This simplifies to:
\(-5x + 42 = 67\)
Subtract 42 from both sides:
\(-5x = 25\)
Dividing both sides by \(-5\) gives:
\[ x = -5 \]
So the solution to the system of equations is:
\(\boxed{(x, y) = (-5, 6)}\)
Thus, the correct option is B: \(x = -5, y = 6\).