Certainly! To solve the system of linear equations:
[tex]\[
\left\{\begin{array}{l}
2x + 5y = 1 \\
2x + y = 5
\end{array}\right.
\][/tex]
we can follow these steps:
1. Label the equations for reference:
[tex]\[
\begin{aligned}
\text{(1)} & \quad 2x + 5y = 1 \\
\text{(2)} & \quad 2x + y = 5
\end{aligned}
\][/tex]
2. Eliminate one variable:
We can eliminate [tex]\(x\)[/tex] by subtracting Equation (2) from Equation (1).
First, let's subtract Equation (2) from Equation (1):
[tex]\[
(2x + 5y) - (2x + y) = 1 - 5
\][/tex]
3. Simplify the resulting equation:
[tex]\[
2x + 5y - 2x - y = -4
\][/tex]
[tex]\[
4y = -4
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-4}{4} = -1
\][/tex]
5. Substitute [tex]\(y = -1\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex]:
Let's use Equation (2):
[tex]\[
2x + y = 5
\][/tex]
Substitute [tex]\(y = -1\)[/tex]:
[tex]\[
2x - 1 = 5
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
2x = 6
\][/tex]
[tex]\[
x = \frac{6}{2} = 3
\][/tex]
So, the solution to the system of equations is:
[tex]\[
x = 3, \quad y = -1
\][/tex]
Therefore, the solution to the given system of equations is [tex]\((x, y) = (3, -1)\)[/tex].
