We are given the following system of simultaneous equations:

1. $5x + y = 21$
2. $x - 3y = 9$

Step 1: Express $y$ in terms of $x$ from the first equation

From the first equation, solve for $y$:

$y = 21 - 5x$

Step 2: Substitute $y = 21 - 5x$ into the second equation

Now, substitute this expression for $y$ into the second equation:

$x - 3(21 - 5x) = 9$

Step 3: Solve for $x$

Expand the equation:

$x - 63 + 15x = 9$

Combine like terms:

$16x - 63 = 9$

Add 63 to both sides:

$16x = 72$

Solve for $x$:

$x = \frac{72}{16} = 4.5$

Step 4: Substitute $x = 4.5$ into the equation for $y$

Now, substitute $x = 4.5$ into $y = 21 - 5x$:

$y = 21 - 5(4.5) = 21 - 22.5 = -1.5$

Final Answer:

The solution to the system of equations is: $x = 4.5, \quad y = -1.5$

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the solution change if the second equation were $x - 3y = 12$?
2. Can simultaneous equations always be solved using substitution?
3. How does the method of elimination compare to substitution in solving simultaneous equations?
4. What is the geometric interpretation of solving simultaneous linear equations?
5. What happens if two equations represent parallel lines?

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