Calculus Examples

$x - 2 y + z = 4$ , $3 x - 5 y - 17 z = 3$ , $2 x - 6 y + 43 z = - 5$

Step 1

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

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Step 1.1

Add $2 y$ to both sides of the equation.

$x + z = 4 + 2 y$

$3 x - 5 y - 17 z = 3$

$2 x - 6 y + 43 z = - 5$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 4 + 2 y - z$

$3 x - 5 y - 17 z = 3$

$2 x - 6 y + 43 z = - 5$

$x = 4 + 2 y - z$

$3 x - 5 y - 17 z = 3$

$2 x - 6 y + 43 z = - 5$

Step 2

Replace all occurrences of $x$ with $4 + 2 y - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x - 5 y - 17 z = 3$ with $4 + 2 y - z$ .

$3 (4 + 2 y - z) - 5 y - 17 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (4 + 2 y - z) - 5 y - 17 z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 \cdot 4 + 3 (2 y) + 3 (- z) - 5 y - 17 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $4$ .

$12 + 3 (2 y) + 3 (- z) - 5 y - 17 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

Step 2.2.1.1.2.2

Multiply $2$ by $3$ .

$12 + 6 y + 3 (- z) - 5 y - 17 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

Step 2.2.1.1.2.3

Multiply $- 1$ by $3$ .

$12 + 6 y - 3 z - 5 y - 17 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

$12 + 6 y - 3 z - 5 y - 17 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

$12 + 6 y - 3 z - 5 y - 17 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $5 y$ from $6 y$ .

$12 + y - 3 z - 17 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

Step 2.2.1.2.2

Subtract $17 z$ from $- 3 z$ .

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$2 x - 6 y + 43 z = - 5$

Step 2.3

Replace all occurrences of $x$ in $2 x - 6 y + 43 z = - 5$ with $4 + 2 y - z$ .

$2 (4 + 2 y - z) - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (4 + 2 y - z) - 6 y + 43 z$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$2 \cdot 4 + 2 (2 y) + 2 (- z) - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $4$ .

$8 + 2 (2 y) + 2 (- z) - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

Step 2.4.1.1.2.2

Multiply $2$ by $2$ .

$8 + 4 y + 2 (- z) - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $2$ .

$8 + 4 y - 2 z - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$8 + 4 y - 2 z - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$8 + 4 y - 2 z - 6 y + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $6 y$ from $4 y$ .

$8 - 2 y - 2 z + 43 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

Step 2.4.1.2.2

Add $- 2 z$ and $43 z$ .

$8 - 2 y + 41 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$8 - 2 y + 41 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$8 - 2 y + 41 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$8 - 2 y + 41 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

$8 - 2 y + 41 z = - 5$

$12 + y - 20 z = 3$

$x = 4 + 2 y - z$

Step 3

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

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Step 3.1

Subtract $12$ from both sides of the equation.

$y - 20 z = 3 - 12$

$8 - 2 y + 41 z = - 5$

$x = 4 + 2 y - z$

Step 3.2

Add $20 z$ to both sides of the equation.

$y = 3 - 12 + 20 z$

$8 - 2 y + 41 z = - 5$

$x = 4 + 2 y - z$

Step 3.3

Subtract $12$ from $3$ .

$y = - 9 + 20 z$

$8 - 2 y + 41 z = - 5$

$x = 4 + 2 y - z$

$y = - 9 + 20 z$

$8 - 2 y + 41 z = - 5$

$x = 4 + 2 y - z$

Step 4

Replace all occurrences of $y$ with $- 9 + 20 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $8 - 2 y + 41 z = - 5$ with $- 9 + 20 z$ .

$8 - 2 (- 9 + 20 z) + 41 z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $8 - 2 (- 9 + 20 z) + 41 z$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$8 - 2 \cdot - 9 - 2 (20 z) + 41 z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

Step 4.2.1.1.2

Multiply $- 2$ by $- 9$ .

$8 + 18 - 2 (20 z) + 41 z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

Step 4.2.1.1.3

Multiply $20$ by $- 2$ .

$8 + 18 - 40 z + 41 z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

$8 + 18 - 40 z + 41 z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $8$ and $18$ .

$26 - 40 z + 41 z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

Step 4.2.1.2.2

Add $- 40 z$ and $41 z$ .

$26 + z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

$26 + z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

$26 + z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

$26 + z = - 5$

$y = - 9 + 20 z$

$x = 4 + 2 y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 4 + 2 y - z$ with $- 9 + 20 z$ .

$x = 4 + 2 (- 9 + 20 z) - z$

$26 + z = - 5$

$y = - 9 + 20 z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $4 + 2 (- 9 + 20 z) - z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$x = 4 + 2 \cdot - 9 + 2 (20 z) - z$

$26 + z = - 5$

$y = - 9 + 20 z$

Step 4.4.1.1.2

Multiply $2$ by $- 9$ .

$x = 4 - 18 + 2 (20 z) - z$

$26 + z = - 5$

$y = - 9 + 20 z$

Step 4.4.1.1.3

Multiply $20$ by $2$ .

$x = 4 - 18 + 40 z - z$

$26 + z = - 5$

$y = - 9 + 20 z$

$x = 4 - 18 + 40 z - z$

$26 + z = - 5$

$y = - 9 + 20 z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $18$ from $4$ .

$x = - 14 + 40 z - z$

$26 + z = - 5$

$y = - 9 + 20 z$

Step 4.4.1.2.2

Subtract $z$ from $40 z$ .

$x = - 14 + 39 z$

$26 + z = - 5$

$y = - 9 + 20 z$

$x = - 14 + 39 z$

$26 + z = - 5$

$y = - 9 + 20 z$

$x = - 14 + 39 z$

$26 + z = - 5$

$y = - 9 + 20 z$

$x = - 14 + 39 z$

$26 + z = - 5$

$y = - 9 + 20 z$

$x = - 14 + 39 z$

$26 + z = - 5$

$y = - 9 + 20 z$

Step 5

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

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Step 5.1

Subtract $26$ from both sides of the equation.

$z = - 5 - 26$

$x = - 14 + 39 z$

$y = - 9 + 20 z$

Step 5.2

Subtract $26$ from $- 5$ .

$z = - 31$

$x = - 14 + 39 z$

$y = - 9 + 20 z$

$z = - 31$

$x = - 14 + 39 z$

$y = - 9 + 20 z$

Step 6

Replace all occurrences of $z$ with $- 31$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = - 14 + 39 z$ with $- 31$ .

$x = - 14 + 39 (- 31)$

$z = - 31$

$y = - 9 + 20 z$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 14 + 39 (- 31)$ .

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Step 6.2.1.1

Multiply $39$ by $- 31$ .

$x = - 14 - 1209$

$z = - 31$

$y = - 9 + 20 z$

Step 6.2.1.2

Subtract $1209$ from $- 14$ .

$x = - 1223$

$z = - 31$

$y = - 9 + 20 z$

$x = - 1223$

$z = - 31$

$y = - 9 + 20 z$

$x = - 1223$

$z = - 31$

$y = - 9 + 20 z$

Step 6.3

Replace all occurrences of $z$ in $y = - 9 + 20 z$ with $- 31$ .

$y = - 9 + 20 (- 31)$

$x = - 1223$

$z = - 31$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 9 + 20 (- 31)$ .

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Step 6.4.1.1

Multiply $20$ by $- 31$ .

$y = - 9 - 620$

$x = - 1223$

$z = - 31$

Step 6.4.1.2

Subtract $620$ from $- 9$ .

$y = - 629$

$x = - 1223$

$z = - 31$

$y = - 629$

$x = - 1223$

$z = - 31$

$y = - 629$

$x = - 1223$

$z = - 31$

$y = - 629$

$x = - 1223$

$z = - 31$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 1223, - 629, - 31)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 1223, - 629, - 31)$

Equation Form:

$x = - 1223, y = - 629, z = - 31$