We are solving the system of linear equations using the elimination method:

$\text{1) } 3x + 5y = 13$ $\text{2) } 2x + y = 4$

Step 1: Make the coefficients of one variable the same

We will eliminate $y$ by making the coefficients of $y$ in both equations equal. To do that, multiply the second equation by 5:

$5(2x + y) = 5(4)$ $\text{This gives: } 10x + 5y = 20 \quad \text{(Equation 3)}$

Now the system is: $\text{1) } 3x + 5y = 13$ $\text{3) } 10x + 5y = 20$

Step 2: Subtract the equations

Subtract equation 1 from equation 3 to eliminate $y$:

$(10x + 5y) - (3x + 5y) = 20 - 13$ $7x = 7$

Step 3: Solve for $x$

$x = \frac{7}{7} = 1$

Step 4: Substitute $x = 1$ into one of the original equations

Substitute $x = 1$ into equation 2: $2(1) + y = 4$ $2 + y = 4$ $y = 4 - 2 = 2$

Final Solution:

$x = 1, \quad y = 2$

The solution to the system of equations is $x = 1$ and $y = 2$.

Would you like further details or have any questions?

Related questions:

1. How does the elimination method differ from the substitution method?
2. What happens if both variables have the same coefficient initially?
3. How can you solve a system where the equations are dependent or inconsistent?
4. Can the elimination method be applied to more than two equations?
5. What if both variables are eliminated and you get a false statement?

Tip: Always check your solution by substituting the values back into the original equations to verify correctness.