Calculus Examples

$x + y - 3 z = 2$ $2 x - y + 2 z = - 21$

Step 1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1

Subtract $x$ from both sides of the equation.

$y - 3 z = 2 - x$

$2 x - y + 2 z = - 21$

Step 1.2

Add $3 z$ to both sides of the equation.

$y = 2 - x + 3 z$

$2 x - y + 2 z = - 21$

$y = 2 - x + 3 z$

$2 x - y + 2 z = - 21$

Step 2

Solve the equation for $z$ .

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Step 2.1

Simplify $2 x - (2 - x + 3 z) + 2 z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$2 x - 1 \cdot 2 + x - (3 z) + 2 z = - 21$

$y = 2 - x + 3 z$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $- 1$ by $2$ .

$2 x - 2 + x - (3 z) + 2 z = - 21$

$y = 2 - x + 3 z$

Step 2.1.1.2.2

Multiply $- - x$ .

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Step 2.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$2 x - 2 + 1 x - (3 z) + 2 z = - 21$

$y = 2 - x + 3 z$

Step 2.1.1.2.2.2

Multiply $x$ by $1$ .

$2 x - 2 + x - (3 z) + 2 z = - 21$

$y = 2 - x + 3 z$

$2 x - 2 + x - (3 z) + 2 z = - 21$

$y = 2 - x + 3 z$

Step 2.1.1.2.3

Multiply $3$ by $- 1$ .

$2 x - 2 + x - 3 z + 2 z = - 21$

$y = 2 - x + 3 z$

$2 x - 2 + x - 3 z + 2 z = - 21$

$y = 2 - x + 3 z$

$2 x - 2 + x - 3 z + 2 z = - 21$

$y = 2 - x + 3 z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Add $2 x$ and $x$ .

$3 x - 2 - 3 z + 2 z = - 21$

$y = 2 - x + 3 z$

Step 2.1.2.2

Add $- 3 z$ and $2 z$ .

$3 x - 2 - z = - 21$

$y = 2 - x + 3 z$

$3 x - 2 - z = - 21$

$y = 2 - x + 3 z$

$3 x - 2 - z = - 21$

$y = 2 - x + 3 z$

Step 2.2

Move all terms not containing $z$ to the right side of the equation.

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Step 2.2.1

Subtract $3 x$ from both sides of the equation.

$- 2 - z = - 21 - 3 x$

$y = 2 - x + 3 z$

Step 2.2.2

Add $2$ to both sides of the equation.

$- z = - 21 - 3 x + 2$

$y = 2 - x + 3 z$

Step 2.2.3

Add $- 21$ and $2$ .

$- z = - 3 x - 19$

$y = 2 - x + 3 z$

$- z = - 3 x - 19$

$y = 2 - x + 3 z$

Step 2.3

Divide each term in $- z = - 3 x - 19$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- z = - 3 x - 19$ by $- 1$ .

$\frac{- z}{- 1} = \frac{- 3 x}{- 1} + \frac{- 19}{- 1}$

$y = 2 - x + 3 z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{z}{1} = \frac{- 3 x}{- 1} + \frac{- 19}{- 1}$

$y = 2 - x + 3 z$

Step 2.3.2.2

Divide $z$ by $1$ .

$z = \frac{- 3 x}{- 1} + \frac{- 19}{- 1}$

$y = 2 - x + 3 z$

$z = \frac{- 3 x}{- 1} + \frac{- 19}{- 1}$

$y = 2 - x + 3 z$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Move the negative one from the denominator of $\frac{- 3 x}{- 1}$ .

$z = - 1 \cdot (- 3 x) + \frac{- 19}{- 1}$

$y = 2 - x + 3 z$

Step 2.3.3.1.2

Rewrite $- 1 \cdot (- 3 x)$ as $- (- 3 x)$ .

$z = - (- 3 x) + \frac{- 19}{- 1}$

$y = 2 - x + 3 z$

Step 2.3.3.1.3

Multiply $- 3$ by $- 1$ .

$z = 3 x + \frac{- 19}{- 1}$

$y = 2 - x + 3 z$

Step 2.3.3.1.4

Divide $- 19$ by $- 1$ .

$z = 3 x + 19$

$y = 2 - x + 3 z$

$z = 3 x + 19$

$y = 2 - x + 3 z$

$z = 3 x + 19$

$y = 2 - x + 3 z$

$z = 3 x + 19$

$y = 2 - x + 3 z$

$z = 3 x + 19$

$y = 2 - x + 3 z$

Step 3

Simplify the right side.

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Step 3.1

Simplify $2 - x + 3 (3 x + 19)$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$y = 2 - x + 3 (3 x) + 3 \cdot 19$

$z = 3 x + 19$

Step 3.1.1.2

Multiply $3$ by $3$ .

$y = 2 - x + 9 x + 3 \cdot 19$

$z = 3 x + 19$

Step 3.1.1.3

Multiply $3$ by $19$ .

$y = 2 - x + 9 x + 57$

$z = 3 x + 19$

$y = 2 - x + 9 x + 57$

$z = 3 x + 19$

Step 3.1.2

Simplify by adding terms.

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Step 3.1.2.1

Add $2$ and $57$ .

$y = - x + 9 x + 59$

$z = 3 x + 19$

Step 3.1.2.2

Add $- x$ and $9 x$ .

$y = 8 x + 59$

$z = 3 x + 19$

$y = 8 x + 59$

$z = 3 x + 19$

$y = 8 x + 59$

$z = 3 x + 19$

$y = 8 x + 59$

$z = 3 x + 19$