To solve the given system of linear equations, let's go through the steps:

The equations in the system are:

$x + 4y = 13$ $-2x + 3y = 18$

Step 1: Solve for $x$ or $y$

Let's use the substitution or elimination method. Here, I will use the elimination method to eliminate $x$.

Step 2: Make the Coefficients of $x$ the Same

Multiply the first equation by 2 to match the coefficient of $x$ in the second equation:

$2(x + 4y) = 2 \cdot 13$ $2x + 8y = 26$

Now, our system of equations is:

$2x + 8y = 26$ $-2x + 3y = 18$

Step 3: Add the Equations

Adding these two equations will eliminate $x$:

$(2x - 2x) + (8y + 3y) = 26 + 18$ $0 + 11y = 44$ $11y = 44$

Step 4: Solve for $y$

Divide both sides by 11:

$y = \frac{44}{11} = 4$

Step 5: Substitute $y = 4$ into the First Equation

Now, substitute $y = 4$ into the first equation to solve for $x$:

$x + 4(4) = 13$ $x + 16 = 13$ $x = 13 - 16$ $x = -3$

Solution

The solution to the system is:

$x = -3, \quad y = 4$

Would you like further details or have any questions?

Here are some related questions for further practice:

1. How would the solution change if the second equation was altered?
2. Can this system be solved graphically, and what would the intersection point represent?
3. What other methods can be used to solve this system, such as substitution or matrix methods?
4. How would the solution process change if the coefficients of $y$ were different?
5. How can you verify that your solution is correct?

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