Question

Solve the following system of equations by the elimination method:

`x/a = y/b; ax + by = a^2 + b^2, a ≠ 0, b ≠ 0`

Answer 1

Given the system of equations:

`x/a = y/b; ax + by = a^2 + b^2, a ≠ 0, b ≠ 0`

Step 1: Express one variable in terms of the other from the first equation

From

`x/a = y/b`

Cross-multiplying:

bx = ay

⇒ `y = b/a x`

Step 2: Substitute `(y = b/a x )` into the second equation

Given:

ax + by = a² + b²

Substituting (y):

`ax + b(b/a x) = a^2 + b^2`

Simplify:

`ax + b^2/a x = a^2 + b^2`

Multiply through by (a) to clear the denominator:

a²x + b²x = a(a² + b²)

Factor (x) on the left:

(a² + b²)x = a(a² + b²)

Step 3: Solve for (x)

Because a² + b² ≠ 0 since a ≠ 0, b ≠ 0,

`x = (a(a^2 + b^2))/(a^2 + b^2)`

x = a

Step 4: Find (y)

Substitute (x = a) into `(y = b/a x )`:

`y = b/a xx a`

y = b

The solution to the system by the elimination method is x = a and y = b.