Solve simultaneous linear equations: (i) x(x + z) – y (y + z)

Solve simultaneous linear equations:

(i) x(x + z) – y (y + z)
(ii) a¹²x⁴ – a⁴x¹²

Solution:

(i) x(x + z) – y (y + z)
x² + xz – y² – yz
Rearrange the above terms we get,
x² – y² + xz – yz
We know that, (a² – b²) = (a + b) (a – b)
So, (x + y) (x – y) + z(x – y)
(x – y) (x + y + z)
(ii) a¹²x⁴ – a⁴x¹²
a¹²x⁴ – a⁴x¹²
Take out common in both terms,
a⁴x⁴ (a⁸ – x⁸)
a⁴x⁴((a⁴)² – (x⁴)²)
We know that, (a² – b²) = (a + b) (a – b)
a⁴x⁴ (a⁴ + x⁴) (a⁴ – x⁴)
a⁴x⁴ (a⁴ + x⁴) ((a²)² – (x²)²)
a⁴x⁴(a⁴ + x⁴) (a² + x²) (a² – x²)
a⁴x⁴ (a⁴ + x⁴) (a² + x²) (a + x) (a – x)

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