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If $| x | + x + y = 10$ and $x + | y | - y = 12$ then find the value of $x + y$ .

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\frac{18}{5}

\frac{17}{5}

\frac{8}{5}

None of these

answer is A.

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Detailed Solution

Given that,

| x | + x + y = 10

and

x + | y | - y = 12

.
Assuming

y > 0

leads to

x = 12

from 2nd equation as:

x + y - y = 12

x = 12

And then from 1st Equation,

|x| + x + y = 10

2 x + y = 10

y = 14

which is contradiction as

y > 0

.
Assuming

y \leq 0

leads to two cases,
(i) If

x \leq 0

then from 1st equation

y = 10

and that is contradictory again.
(ii) If

x > 0

then

2 x + y = 10 \dots (1),

x - 2 y = 12 \dots (2)

Multiplying equation (1) by 2 and adding to equation (2) we get,

4 x + 2 y + x - 2 y = 20 + 12

5 x = 32

x = \frac{32}{5}

Substituting the value of

x

in equation (2),

\frac{32}{5} - 2 y = 12

2 y = \frac{32}{5} - 12

y = \frac{- 14}{5}

∴ x + y = \frac{32}{5} + \frac{- 14}{5}

∴ x + y = \frac{18}{5}

Therefore, option (1) is correct.

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