To solve the given system of linear equations:
[tex]\[
\begin{cases}
4x + y = 16 \\
2x + 3y = -2
\end{cases}
\][/tex]
we can use the method of substitution or elimination, but here we'll use elimination for a straightforward solution.
1. Express the system of equations:
[tex]\[
\begin{cases}
4x + y = 16 \quad \text{(1)} \\
2x + 3y = -2 \quad \text{(2)}
\end{cases}
\][/tex]
2. Multiply equation (2) by 2 to align the coefficients of [tex]\( x \)[/tex]:
[tex]\[
4x + y = 16 \quad \text{(1)}
\][/tex]
[tex]\[
4x + 6y = -4 \quad \text{(3)}
\][/tex]
3. Subtract equation (1) from equation (3):
[tex]\[
(4x + 6y) - (4x + y) = -4 - 16
\][/tex]
Simplify the left-hand side and the right-hand side:
[tex]\[
4x + 6y - 4x - y = -20
\][/tex]
[tex]\[
5y = -20
\][/tex]
Now, solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-20}{5} = -4
\][/tex]
4. Substitute [tex]\( y = -4 \)[/tex] back into equation (1) to find [tex]\( x \)[/tex]:
[tex]\[
4x + (-4) = 16
\][/tex]
Simplify:
[tex]\[
4x - 4 = 16
\][/tex]
Add 4 to both sides:
[tex]\[
4x = 20
\][/tex]
Divide by 4:
[tex]\[
x = \frac{20}{4} = 5
\][/tex]
5. Conclusion:
The solution to the system of equations is [tex]\( (x, y) = (5, -4) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{(5, -4)} \][/tex] which corresponds to option A.
