To solve the given system of equations using the linear combination (or elimination) method, follow these steps:
Given the system:
[tex]\[
\begin{cases}
5x + y = 2 \\
4x + y = 5
\end{cases}
\][/tex]
1. Formulate the system of equations:
[tex]\[
\begin{aligned}
1. & \quad 5x + y = 2 \quad \text{(Equation 1)} \\
2. & \quad 4x + y = 5 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
2. Eliminate one variable (y):
- Subtract Equation 2 from Equation 1 to eliminate [tex]\( y \)[/tex].
[tex]\[
(5x + y) - (4x + y) = 2 - 5
\][/tex]
3. Simplify the resulting equation to solve for [tex]\( x \)[/tex]:
[tex]\[
5x + y - 4x - y = 2 - 5
\][/tex]
[tex]\[
x = -3
\][/tex]
4. Substitute [tex]\( x = -3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
- Using Equation 1:
[tex]\[
5(-3) + y = 2
\][/tex]
[tex]\[
-15 + y = 2
\][/tex]
[tex]\[
y = 2 + 15
\][/tex]
[tex]\[
y = 17
\][/tex]
Hence, the solution to the system of equations is:
[tex]\[
(x, y) = (-3, 17)
\][/tex]
This matches the solution provided in the multiple-choice options:
[tex]\[
(-3, 17)
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{(-3, 17)}
\][/tex]
