To solve the system of simultaneous equations:

4x + y = 0 \tag{1} x - y = 5 \tag{2}

Step 1: Solve one equation for one variable

From equation $(2)$, solve for $x$:

$x = y + 5$

Step 2: Substitute into the other equation

Substitute $x = y + 5$ into equation $(1)$:

$4(y + 5) + y = 0$

Expand the expression:

$4y + 20 + y = 0$

Simplify:

$5y + 20 = 0$

Step 3: Solve for $y$

Subtract 20 from both sides:

$5y = -20$

Now, divide by 5:

$y = -4$

Step 4: Substitute $y = -4$ back into the expression for $x$

Substitute $y = -4$ into $x = y + 5$:

$x = -4 + 5 = 1$

Final solution:

$x = 1, \quad y = -4$

The solution to the simultaneous equations is: $x = 1, \quad y = -4$

Would you like further details on this solution?

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Here are 8 related questions for further exploration:

1. How would you solve the equations if they were nonlinear?
2. What happens if you change the second equation to $x + y = 5$?
3. How can simultaneous equations be solved graphically?
4. Can you solve the system using matrix methods like Cramer's Rule?
5. How would the solution change if $4x + y = 2$?
6. What is the determinant of the system if written as a matrix?
7. How can you verify the solution by substituting into both equations?
8. How can simultaneous equations be applied in real-world problems?

Tip: Always check your solutions by substituting the values of the variables back into the original equations.