If $3x + 2|y|+ y = 7$ and $x + |x|+ 3y = 1$, then $x + 2y$ is

We have two equations involving absolute values:

$3x + 2|y| + y = 7$ ... (1)

$x + |x| + 3y = 1$ ... (2)

When we see absolute values like $|x|$ and $|y|$, we need to consider different cases based on whether these variables are positive or negative.

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The absolute value function behaves differently depending on the sign of the variable:

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

If $x < 0$, then $|x| = -x$

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

If $y < 0$, then $|y| = -y$

Since we have both $|x|$ and $|y|$, we need to check all possible sign combinations.

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

$|x| = x$ and $|y| = y$

Our equations become:

$3x + 2y + y = 7$

$3x + 3y = 7$ ... (1')

$x + x + 3y = 1$

$2x + 3y = 1$ ... (2')

$(3x + 3y) - (2x + 3y) = 7 - 1$

$x = 6$

$2(6) + 3y = 1$

$12 + 3y = 1$

$y = -\frac{11}{3}$

We assumed $y \geq 0$, but we got $y = -\frac{11}{3} < 0$.

This case is invalid.

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

$|x| = x$ and $|y| = -y$

Our equations become:

$3x + 2(-y) + y = 7$

$3x - 2y + y = 7$

$3x - y = 7$ ... (1')

$x + x + 3y = 1$

$2x + 3y = 1$ ... (2')

From equation (1'): $y = 3x - 7$

$2x + 3(3x - 7) = 1$

$2x + 9x - 21 = 1$

$11x = 22$

$x = 2$

$y = 3(2) - 7 = -1$

$x = 2 \geq 0$ and $y = -1 < 0$

This case is valid.

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

$|x| = -x$ and $|y| = y$

Our equations become:

$3x + 2y + y = 7$

$3x + 3y = 7$ ... (1')

$x + (-x) + 3y = 1$

$3y = 1$ ... (2')

From equation (2'): $y = \frac{1}{3}$

$3x + 3(\frac{1}{3}) = 7$

$3x + 1 = 7$

$x = 2$

We assumed $x < 0$, but we got $x = 2 > 0$.

This case is invalid.

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

$|x| = -x$ and $|y| = -y$

Our equations become:

$3x + 2(-y) + y = 7$

$3x - y = 7$ ... (1')

$x + (-x) + 3y = 1$

$3y = 1$ ... (2')

From equation (2'): $y = \frac{1}{3}$

We assumed $y < 0$, but we got $y = \frac{1}{3} > 0$.

This case is invalid.

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Only Case II gives us a valid solution that satisfies both the equations and the sign assumptions.

$x = 2$ and $y = -1$

Therefore: $x + 2y = 2 + 2(-1) = 2 - 2 = 0$