If x and y are real numbers such that $|x| + x + y = 15$ and $x + |y| - y = 20$, then $(x - y)$ equals

Given:

$|x| + x + y = 15$ ... (equation 1)

$x + |y| - y = 20$ ... (equation 2)

Understanding Absolute Values:

Remember that absolute value removes the negative sign:

If a number is positive or zero: $|a| = a$

If a number is negative: $|a| = -a$

Since we don't know if $x$ and $y$ are positive or negative, we need to consider all possibilities. This is because the absolute value expressions $|x|$ and $|y|$ behave differently depending on the sign of the variable.

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Smart Observation Before Starting Cases:

From equation 1: $|x| + x = 15 - y$

From equation 2: $|y| - y = 20 - x$

Key Insight:

If $x \geq 0$, then $|x| + x = x + x = 2x$

If $x < 0$, then $|x| + x = (-x) + x = 0$

If $y \geq 0$, then $|y| - y = y - y = 0$

If $y < 0$, then $|y| - y = (-y) - y = -2y$

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Case 1: $x \geq 0$ and $y \geq 0$

Using our insights: $|x| + x = 2x$ and $|y| - y = 0$

Our equations become:

$2x = 15 - y$ ... (A)

$0 = 20 - x$

$\rightarrow x = 20$ ... (B)

Substituting $x = 20$ into equation (A):

$2(20) = 15 - y$

$40 = 15 - y$

$y = -25$

Check: We assumed $y \geq 0$, but got $y = -25 < 0$. This contradicts our assumption.

Conclusion: Case 1 is impossible.

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Case 2: $x \geq 0$ and $y < 0$

Using our insights: $|x| + x = 2x$ and $|y| - y = -2y$

Our equations become:

$2x = 15 - y$ ... (A)

$-2y = 20 - x$ ... (B)

From equation (B): $x = 20 + 2y$

Substituting into equation (A):

$2(20 + 2y) = 15 - y$

$40 + 4y = 15 - y$

$5y = -25$

$y = -5$

Finding $x$: $x = 20 + 2(-5) = 20 - 10 = 10$

Check: $x = 10 \geq 0$ and $y = -5 < 0$

Conclusion: Case 2 gives us a valid solution: $x = 10, y = -5$

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Case 3: $x < 0$ and $y \geq 0$

Using our insights: $|x| + x = 0$ and $|y| - y = 0$

Our equations become:

$0 = 15 - y$

$\rightarrow y = 15$

$0 = 20 - x$

$\rightarrow x = 20$

Check: We assumed $x < 0$, but got $x = 20 > 0$. This contradicts our assumption.

Conclusion: Case 3 is impossible.

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Case 4: $x < 0$ and $y < 0$

Using our insights: $|x| + x = 0$ and $|y| - y = -2y$

Our equations become:

$0 = 15 - y$

$\rightarrow y = 15$

$-2y = 20 - x$

From the first equation: $y = 15$

From the second equation: $-2(15) = 20 - x$

$-30 = 20 - x$

$x = 50$

Check: We assumed both $x < 0$ and $y < 0$, but got $x = 50 > 0$ and $y = 15 > 0$. This contradicts our assumptions.

Conclusion: Case 4 is impossible.

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Final Answer:

Only Case 2 gives us a valid solution: $x = 10$ and $y = -5$

Therefore: $x - y = 10 - (-5) = 10 + 5 = 15$