To solve the simultaneous equations:

x + 2y = 13 \tag{1} 3x + y = 24 \tag{2}

Step 1: Solve one equation for one variable

From equation (1), express $x$ in terms of $y$: x = 13 - 2y \tag{3}

Step 2: Substitute into the second equation

Substitute $x = 13 - 2y$ from (3) into equation (2): $3(13 - 2y) + y = 24$

Simplify: $39 - 6y + y = 24$ $39 - 5y = 24$

Rearrange to solve for $y$: $-5y = 24 - 39$ $-5y = -15$ $y = 3$

Step 3: Substitute $y = 3$ back into equation (3)

Substitute $y = 3$ into $x = 13 - 2y$: $x = 13 - 2(3)$ $x = 13 - 6$ $x = 7$

Final Solution:

$x = 7, \, y = 3$

Would you like a more detailed breakdown of any of the steps? Here are some related questions:

1. What are other methods for solving simultaneous equations (e.g., substitution vs. elimination)?
2. Can simultaneous equations always have a unique solution?
3. How can these equations be visualized graphically?
4. What happens if the equations describe parallel lines?
5. How do we approach systems with more than two equations?

Tip: Always check your solution by substituting the values back into the original equations!