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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.

Work out x and y in terms of a and b by solving the simultaneous equations below. Fully simplify your answer. y = ax + b, x = ay + b, where a and b are numbers greater than 1.

Learn how to solve simultaneous equations for x and y in terms of a and b. This problem demonstrates the substitution method, leading to a simplified expression for both variables. Suitable for students in Grades 10-12.

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