To solve the system of equations using the elimination method, we have the following equations:
- \( 2x - 3y = 14 \)
- \( 4x + 3y = 10 \)
We can eliminate \( y \) by adding the two equations together. First, let's manipulate the first equation to make the coefficients of \( y \) in both equations match in magnitude but opposite in sign.
Multiply the first equation by 1, and leave the second as is:
- \( 2x - 3y = 14 \)
- \( 4x + 3y = 10 \)
Now, let's add both equations:
\[ (2x - 3y) + (4x + 3y) = 14 + 10 \]
This simplifies to:
\[ (2x + 4x) + (-3y + 3y) = 24 \]
\[ 6x = 24 \]
Now, solve for \( x \):
\[ x = \frac{24}{6} = 4 \]
Next, substitute \( x = 4 \) back into one of the original equations to find \( y \). We'll use the first equation:
\[ 2(4) - 3y = 14 \]
This simplifies to:
\[ 8 - 3y = 14 \]
Now isolate \( y \):
\[ -3y = 14 - 8 \] \[ -3y = 6 \] \[ y = \frac{6}{-3} = -2 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (4, -2) \]
So, the final answer is:
\[ (4, -2) \]