To determine the solution to the given system of equations:
[tex]\[
\begin{cases}
7x + 3y = -11 \\
2x + 5y = 1
\end{cases}
\][/tex]
we will follow these steps:
1. Write down the system of linear equations:
[tex]\[
7x + 3y = -11 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
2x + 5y = 1 \quad \text{(Equation 2)}
\][/tex]
2. Choose one of the substitution or elimination methods to solve the system. In this case, let's use the elimination method to eliminate [tex]\( y \)[/tex].
3. Multiply Equation 1 by 5 and Equation 2 by 3 so that the coefficients of [tex]\( y \)[/tex] will be the same:
[tex]\[
\begin{cases}
5 \cdot (7x + 3y) = 5 \cdot (-11) \\
3 \cdot (2x + 5y) = 3 \cdot 1
\end{cases}
\][/tex]
which simplifies to:
[tex]\[
\begin{cases}
35x + 15y = -55 \\
6x + 15y = 3
\end{cases}
\][/tex]
4. Subtract the second equation from the first to eliminate [tex]\( y \)[/tex]:
[tex]\[
(35x + 15y) - (6x + 15y) = -55 - 3
\][/tex]
This simplifies to:
[tex]\[
29x = -58
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-58}{29} = -2
\][/tex]
6. Substitute [tex]\( x = -2 \)[/tex] back into one of the original equations (let's use Equation 2) to find [tex]\( y \)[/tex]:
[tex]\[
2(-2) + 5y = 1
\][/tex]
This simplifies to:
[tex]\[
-4 + 5y = 1
\][/tex]
[tex]\[
5y = 5
\][/tex]
[tex]\[
y = 1
\][/tex]
Thus, the solution to the system of equations is [tex]\( (x, y) = (-2, 1) \)[/tex].
To check our solution against the provided choices:
- [tex]\( (3, -1) \)[/tex]
- [tex]\( (2, 1) \)[/tex]
- [tex]\( (-2, 1) \)[/tex]
- [tex]\( (1, -2) \)[/tex]
We observe that the correct solution [tex]\( (-2, 1) \)[/tex] matches one of the choices provided. Therefore, the solution to the system of equations is:
[tex]\[ \boxed{(-2, 1)} \][/tex]
