The problem consists of a system of linear equations:

1. $2x + 3y = 7$
2. $4x + 5y = 13$

Solution:

We solve this system using the substitution or elimination method. I'll use the elimination method here:

Step 1: Eliminate one variable

To eliminate $x$, make the coefficients of $x$ equal by multiplying the first equation by $2$:

$2(2x + 3y) = 2(7) \quad \Rightarrow \quad 4x + 6y = 14$

The new system is: $4x + 6y = 14$ $4x + 5y = 13$

Subtract the second equation from the first: $(4x + 6y) - (4x + 5y) = 14 - 13$ $y = 1$

Step 2: Solve for $x$

Substitute $y = 1$ into the first equation: $2x + 3(1) = 7$ $2x + 3 = 7$ $2x = 4 \quad \Rightarrow \quad x = 2$

Final Solution:

$x = 2, \, y = 1$

Would you like a detailed explanation or a graph of the solution?

Here are 5 related questions you might explore:

1. How can the substitution method be applied to the same problem?
2. What is the graphical interpretation of this solution?
3. How do you solve systems of equations with more variables?
4. What are the conditions for a system of equations to have no solution or infinitely many solutions?
5. How does the determinant of a matrix relate to solving linear equations?

Tip: Always double-check your solution by substituting the values of $x$ and $y$ back into the original equations.