Solve the following equations :
$x + y + z + u = 2 a, x + y - z - u = 2 b, x - y + z - u = 2 c, x - y - z + u = 2 d .$

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Solution

$x + y + z + u = 2 a, x + y - z - u = 2 b,$ .........(1)

$x - y + z - u = 2 c, x - y - z + u = 2 d$ ...................(2)

Adding first two and adding last two,
we get
$x + y = a + b, x - y = c + d$

$∴ x = \frac{1}{2} (a + b + c + d), y = \frac{1}{2} (a + b - c - d)$

Subtracting first two and subtracting last two,

$z + u = a - b$ and $z - u = c - d$ .

$∴ z = \frac{1}{2} (a - b + c - d), u = \frac{1}{2} (a - b - c + d)$ .

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