Calculus Examples

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

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x - 3 z = 2 - y$

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

Step 1.2

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

$x = 2 - y + 3 z$

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

$x = 2 - y + 3 z$

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

Step 2

Solve the equation for $y$ .

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

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

$x = 2 - y + 3 z$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $2$ by $2$ .

$4 + 2 (- y) + 2 (3 z) - y + 2 z = - 21$

$x = 2 - y + 3 z$

Step 2.1.1.2.2

Multiply $- 1$ by $2$ .

$4 - 2 y + 2 (3 z) - y + 2 z = - 21$

$x = 2 - y + 3 z$

Step 2.1.1.2.3

Multiply $3$ by $2$ .

$4 - 2 y + 6 z - y + 2 z = - 21$

$x = 2 - y + 3 z$

$4 - 2 y + 6 z - y + 2 z = - 21$

$x = 2 - y + 3 z$

$4 - 2 y + 6 z - y + 2 z = - 21$

$x = 2 - y + 3 z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Subtract $y$ from $- 2 y$ .

$4 - 3 y + 6 z + 2 z = - 21$

$x = 2 - y + 3 z$

Step 2.1.2.2

Add $6 z$ and $2 z$ .

$4 - 3 y + 8 z = - 21$

$x = 2 - y + 3 z$

$4 - 3 y + 8 z = - 21$

$x = 2 - y + 3 z$

$4 - 3 y + 8 z = - 21$

$x = 2 - y + 3 z$

Step 2.2

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

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Step 2.2.1

Subtract $4$ from both sides of the equation.

$- 3 y + 8 z = - 21 - 4$

$x = 2 - y + 3 z$

Step 2.2.2

Subtract $8 z$ from both sides of the equation.

$- 3 y = - 21 - 4 - 8 z$

$x = 2 - y + 3 z$

Step 2.2.3

Subtract $4$ from $- 21$ .

$- 3 y = - 25 - 8 z$

$x = 2 - y + 3 z$

$- 3 y = - 25 - 8 z$

$x = 2 - y + 3 z$

Step 2.3

Divide each term in $- 3 y = - 25 - 8 z$ by $- 3$ and simplify.

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Step 2.3.1

Divide each term in $- 3 y = - 25 - 8 z$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{- 25}{- 3} + \frac{- 8 z}{- 3}$

$x = 2 - y + 3 z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{- 25}{- 3} + \frac{- 8 z}{- 3}$

$x = 2 - y + 3 z$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 25}{- 3} + \frac{- 8 z}{- 3}$

$x = 2 - y + 3 z$

$y = \frac{- 25}{- 3} + \frac{- 8 z}{- 3}$

$x = 2 - y + 3 z$

$y = \frac{- 25}{- 3} + \frac{- 8 z}{- 3}$

$x = 2 - y + 3 z$

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

Dividing two negative values results in a positive value.

$y = \frac{25}{3} + \frac{- 8 z}{- 3}$

$x = 2 - y + 3 z$

Step 2.3.3.1.2

Dividing two negative values results in a positive value.

$y = \frac{25}{3} + \frac{8 z}{3}$

$x = 2 - y + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

$x = 2 - y + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

$x = 2 - y + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

$x = 2 - y + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

$x = 2 - y + 3 z$

Step 3

Simplify the right side.

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Step 3.1

Simplify $2 - (\frac{25}{3} + \frac{8 z}{3}) + 3 z$ .

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Step 3.1.1

Apply the distributive property.

$x = 2 - \frac{25}{3} - \frac{8 z}{3} + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 \cdot \frac{3}{3} - \frac{25}{3} - \frac{8 z}{3} + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.3

Combine $2$ and $\frac{3}{3}$ .

$x = \frac{2 \cdot 3}{3} - \frac{25}{3} - \frac{8 z}{3} + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.4

Combine the numerators over the common denominator.

$x = \frac{2 \cdot 3 - 25}{3} - \frac{8 z}{3} + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $2$ by $3$ .

$x = \frac{6 - 25}{3} - \frac{8 z}{3} + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.5.2

Subtract $25$ from $6$ .

$x = \frac{- 19}{3} - \frac{8 z}{3} + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

$x = \frac{- 19}{3} - \frac{8 z}{3} + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.6

Move the negative in front of the fraction.

$x = - \frac{19}{3} - \frac{8 z}{3} + 3 z$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.7

To write $3 z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{19}{3} - \frac{8 z}{3} + 3 z \cdot \frac{3}{3}$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.8

Combine $3 z$ and $\frac{3}{3}$ .

$x = - \frac{19}{3} - \frac{8 z}{3} + \frac{3 z \cdot 3}{3}$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.9

Combine the numerators over the common denominator.

$x = - \frac{19}{3} + \frac{- 8 z + 3 z \cdot 3}{3}$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.10

Combine the numerators over the common denominator.

$x = \frac{- 19 - 8 z + 3 z \cdot 3}{3}$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.11

Multiply $3$ by $3$ .

$x = \frac{- 19 - 8 z + 9 z}{3}$

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.12

Add $- 8 z$ and $9 z$ .

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

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.13

Rewrite $- 19$ as $- 1 (19)$ .

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

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.14

Factor $- 1$ out of $z$ .

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

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.15

Factor $- 1$ out of $- 1 (19) - 1 (- z)$ .

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

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 3.1.16

Move the negative in front of the fraction.

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

$y = \frac{25}{3} + \frac{8 z}{3}$

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

$y = \frac{25}{3} + \frac{8 z}{3}$

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

$y = \frac{25}{3} + \frac{8 z}{3}$

Step 4

Simplify the right side.

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Step 4.1

Reorder $\frac{25}{3}$ and $\frac{8 z}{3}$ .

$y = \frac{8 z}{3} + \frac{25}{3}$

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

$y = \frac{8 z}{3} + \frac{25}{3}$

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