We are tasked with solving the system of two linear equations in the image. The equations are:

$(a+b)x + (a-b)y = 2a$ $(a-b)x + (a+b)y = 2b$

Let's solve this system step by step.

Step 1: Write the equations

1. Equation 1:
   $(a+b)x + (a-b)y = 2a$

2. Equation 2:
   $(a-b)x + (a+b)y = 2b$

Step 2: Eliminate one variable

We will eliminate $y$ by adding the two equations.

Add Equation 1 and Equation 2:

[ [(a+b) + (a-b)]x + [(a-b) + (a+b)]y = 2a + 2b ]

Simplify terms: $2ax + 2ay = 2a + 2b$

Factor out $2$: $2a(x + y) = 2(a + b)$

Divide through by 2: $a(x + y) = a + b$

Solve for $x + y$: x + y = 1 + \frac{b}{a} \quad \text{(if a \neq 0)}.

Step 3: Eliminate $x$

Now subtract Equation 2 from Equation 1:

[ [(a+b) - (a-b)]x + [(a-b) - (a+b)]y = 2a - 2b ]

Simplify terms: $2bx - 2by = 2a - 2b$

Factor out $2$: $2b(x - y) = 2(a - b)$

Divide through by 2: $b(x - y) = a - b$

Solve for $x - y$: x - y = \frac{a - b}{b} \quad \text{(if b \neq 0)}.

Step 4: Solve for $x$ and $y$

Now we have two equations:

1. $x + y = 1 + \frac{b}{a}$
2. $x - y = \frac{a - b}{b}$

Add these equations to solve for $x$: $(x + y) + (x - y) = \left(1 + \frac{b}{a}\right) + \frac{a - b}{b}$

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

Simplify the terms: $2x = 1 + \frac{b}{a} + \frac{a}{b} - 1$

$2x = \frac{b}{a} + \frac{a}{b}$

$x = \frac{1}{2} \left( \frac{b}{a} + \frac{a}{b} \right)$

Solve for $y$ using $x + y = 1 + \frac{b}{a}$: $y = \left(1 + \frac{b}{a}\right) - x$

Substitute $x = \frac{1}{2} \left( \frac{b}{a} + \frac{a}{b} \right)$ into this expression: $y = 1 + \frac{b}{a} - \frac{1}{2} \left( \frac{b}{a} + \frac{a}{b} \right)$

Simplify: $y = 1 + \frac{b}{a} - \frac{1}{2} \frac{b}{a} - \frac{1}{2} \frac{a}{b}$

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

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

Final Solution:

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

Let me know if you'd like a detailed verification or further assistance!

Related Questions:

1. How can we verify the solution by substituting back into the original equations?
2. What happens if $a = 0$ or $b = 0$?
3. Can we solve the system using a matrix or determinant method?
4. How does the solution simplify for specific values of $a$ and $b$?
5. Can we derive a geometric interpretation for these equations?

Tip:

To check your solution, always substitute $x$ and $y$ back into both original equations!