Calculus Examples

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

Step 1

Rewrite the equation as $x (1 - y) + y (1 - x) = z$ .

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

Step 2

Simplify $x (1 - y) + y (1 - x)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$x \cdot 1 + x (- y) + y (1 - x) = z$

Step 2.1.2

Multiply $x$ by $1$ .

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

Step 2.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.1.4

Apply the distributive property.

$x - x y + y \cdot 1 + y (- x) = z$

Step 2.1.5

Multiply $y$ by $1$ .

$x - x y + y + y (- x) = z$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$x - x y + y - y x = z$

Step 2.2

Subtract $y x$ from $- x y$ .

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Step 2.2.1

Move $y$ .

$x - x y - x y + y = z$

Step 2.2.2

Subtract $x y$ from $- x y$ .

$x - 2 x y + y = z$

Step 3

Subtract $x$ from both sides of the equation.

$- 2 x y + y = z - x$

Step 4

Factor $y$ out of $- 2 x y + y$ .

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Step 4.1

Factor $y$ out of $- 2 x y$ .

$y (- 2 x) + y = z - x$

Step 4.2

Raise $y$ to the power of $1$ .

$y (- 2 x) + y = z - x$

Step 4.3

Factor $y$ out of $y^{1}$ .

$y (- 2 x) + y \cdot 1 = z - x$

Step 4.4

Factor $y$ out of $y (- 2 x) + y \cdot 1$ .

$y (- 2 x + 1) = z - x$

Step 5

Divide each term in $y (- 2 x + 1) = z - x$ by $- 2 x + 1$ and simplify.

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Step 5.1

Divide each term in $y (- 2 x + 1) = z - x$ by $- 2 x + 1$ .

$\frac{y (- 2 x + 1)}{- 2 x + 1} = \frac{z}{- 2 x + 1} + \frac{- x}{- 2 x + 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 2 x + 1$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{y (- 2 x + 1)}{- 2 x + 1} = \frac{z}{- 2 x + 1} + \frac{- x}{- 2 x + 1}$

Step 5.2.1.2

Divide $y$ by $1$ .

$y = \frac{z}{- 2 x + 1} + \frac{- x}{- 2 x + 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Combine the numerators over the common denominator.

$y = \frac{z - x}{- 2 x + 1}$

Step 5.3.2

Factor $- 1$ out of $- 2 x$ .

$y = \frac{z - x}{- (2 x) + 1}$

Step 5.3.3

Rewrite $1$ as $- 1 (- 1)$ .

$y = \frac{z - x}{- (2 x) - 1 (- 1)}$

Step 5.3.4

Factor $- 1$ out of $- (2 x) - 1 (- 1)$ .

$y = \frac{z - x}{- (2 x - 1)}$

Step 5.3.5

Rewrite negatives.

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Step 5.3.5.1

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

$y = \frac{z - x}{- 1 (2 x - 1)}$

Step 5.3.5.2

Move the negative in front of the fraction.

$y = - \frac{z - x}{2 x - 1}$