We are given the following system of simultaneous equations:

$y = ax + b$ $x = ay + b$

We need to solve for $x$ and $y$ in terms of $a$ and $b$.

Step 1: Substitute one equation into the other

From the first equation, solve for $y$ in terms of $x$:

$y = ax + b$

Substitute this expression for $y$ into the second equation:

$x = a(ax + b) + b$

Step 2: Expand and simplify

Now expand the right-hand side:

$x = a^2x + ab + b$

Simplify further:

$x = a^2x + b(a + 1)$

Step 3: Solve for $x$

Rearrange the equation to isolate terms involving $x$ on one side:

$x - a^2x = b(a + 1)$

Factor out $x$:

$x(1 - a^2) = b(a + 1)$

Now, solve for $x$:

$x = \frac{b(a + 1)}{1 - a^2}$

Step 4: Solve for $y$

Now that we have $x$, substitute this back into the equation $y = ax + b$:

$y = a\left(\frac{b(a + 1)}{1 - a^2}\right) + b$

Simplify this expression:

$y = \frac{ab(a + 1)}{1 - a^2} + b$

To combine the terms, express $b$ as:

$y = \frac{ab(a + 1) + b(1 - a^2)}{1 - a^2}$

Simplify the numerator:

$ab(a + 1) + b(1 - a^2) = ab^2 + ab - ab^2 + b = b(a + 1)$

Thus, we get:

$y = \frac{b(a + 1)}{1 - a^2}$

Final Answer:

Both $x$ and $y$ are equal:

$x = y = \frac{b(a + 1)}{1 - a^2}$

Would you like more details, or do you have any further questions?

Here are 5 related questions for practice:

1. What would the solution be if $a = 2$ and $b = 3$?
2. How does the solution change if $b = 0$?
3. Can this system of equations be solved graphically? What would the graphs look like?
4. How would you approach the problem if $a$ or $b$ were negative numbers?
5. What happens to the solution as $a \to 1$?

Tip: To simplify simultaneous equations, substitution is a powerful method, especially when one variable can be easily expressed in terms of the other.