To solve the system of linear equations:
[tex]\[
\begin{cases}
2x + y = 3 \\
3x + 5y = 1
\end{cases}
\][/tex]
we can use the method of elimination or substitution. Here's a step-by-step solution using the elimination method:
1. Write the equations in standard form:
[tex]\[
2x + y = 3 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
3x + 5y = 1 \quad \text{(Equation 2)}
\][/tex]
2. Eliminate one variable by making their coefficients equal:
To eliminate [tex]\( y \)[/tex], we need the coefficients of [tex]\( y \)[/tex] to be the same. We can multiply Equation 1 by 5:
[tex]\[
5(2x + y) = 5(3)
\][/tex]
Simplifying this, we get:
[tex]\[
10x + 5y = 15 \quad \text{(Equation 3)}
\][/tex]
3. Subtract Equation 2 from Equation 3:
[tex]\[
(10x + 5y) - (3x + 5y) = 15 - 1
\][/tex]
Simplifying this, we get:
[tex]\[
7x = 14
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{14}{7} = 2
\][/tex]
5. Substitute [tex]\( x = 2 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]:
Using Equation 1:
[tex]\[
2(2) + y = 3
\][/tex]
Simplifying this, we get:
[tex]\[
4 + y = 3
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = 3 - 4 = -1
\][/tex]
6. Solution:
Hence, the solution to the system of equations is:
[tex]\[
x = 2, \quad y = -1
\][/tex]
So the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the given system of equations are [tex]\( x = 2 \)[/tex] and [tex]\( y = -1 \)[/tex].
