If $\left|x\right|-x+y=42$ and $x+\left|y\right|+y=24$, then what is the value of $x+y$? Express your answer in simplest terms.

$\text{(A) }-4\qquad\text{(B) }\frac{26}{5}\qquad\text{(C) }6\qquad\text{(D) }10\qquad\text{(E) }18$
1. We start with the given equations:
   \[
   \left|x\right| - x + y = 42
   \]
   \[
   x + \left|y\right| + y = 24
   \]

2. Consider the case where \( x \geq 0 \):
   - In this case, \(\left|x\right| = x\), so the first equation becomes:
     \[
     x - x + y = 42 \implies y = 42
     \]
   - Substitute \( y = 42 \) into the second equation:
     \[
     x + \left|42\right| + 42 = 24 \implies x + 42 + 42 = 24 \implies x + 84 = 24 \implies x = 24 - 84 \implies x = -60
     \]
   - Since \( x = -60 \) contradicts our assumption that \( x \geq 0 \), \( x \) must be negative.

3. Consider the case where \( x < 0 \):
   - In this case, \(\left|x\right| = -x\), so the first equation becomes:
     \[
     -x - x + y = 42 \implies -2x + y = 42
     \]
   - The second equation remains:
     \[
     x + \left|y\right| + y = 24
     \]

4. Consider the case where \( y \geq 0 \):
   - In this case, \(\left|y\right| = y\), so the second equation becomes:
     \[
     x + y + y = 24 \implies x + 2y = 24
     \]

5. We now have the system of linear equations:
   \[
   -2x + y = 42
   \]
   \[
   x + 2y = 24
   \]

6. Solve the system of equations:
   - Multiply the second equation by 2:
     \[
     2(x + 2y) = 2 \cdot 24 \implies 2x + 4y = 48
     \]
   - Add the modified second equation to the first equation:
     \[
     -2x + y + 2x + 4y = 42 + 48 \implies 5y = 90 \implies y = 18
     \]
   - Substitute \( y = 18 \) back into the second equation:
     \[
     x + 2(18) = 24 \implies x + 36 = 24 \implies x = 24 - 36 \implies x = -12
     \]

7. Calculate \( x + y \):
   \[
   x + y = -12 + 18 = 6
   \]

The final answer is \(\boxed{6}\).