Algebra Examples

$x - 2 y + 3 z = 9$ $- x + 3 y - z = - 6$ $2 x - 5 y + 5 z = 17$

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.

$x + 3 z = 9 + 2 y$

$- x + 3 y - z = - 6$

$2 x - 5 y + 5 z = 17$

Step 1.2

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

$x = 9 + 2 y - 3 z$

$- x + 3 y - z = - 6$

$2 x - 5 y + 5 z = 17$

$x = 9 + 2 y - 3 z$

$- x + 3 y - z = - 6$

$2 x - 5 y + 5 z = 17$

Step 2

Replace all occurrences of $x$ with $9 + 2 y - 3 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- x + 3 y - z = - 6$ with $9 + 2 y - 3 z$ .

$- (9 + 2 y - 3 z) + 3 y - z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- (9 + 2 y - 3 z) + 3 y - z$ .

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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.

$- 1 \cdot 9 - (2 y) - (- 3 z) + 3 y - z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $9$ .

$- 9 - (2 y) - (- 3 z) + 3 y - z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

Step 2.2.1.1.2.2

Multiply $2$ by $- 1$ .

$- 9 - 2 y - (- 3 z) + 3 y - z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

Step 2.2.1.1.2.3

Multiply $- 3$ by $- 1$ .

$- 9 - 2 y + 3 z + 3 y - z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

$- 9 - 2 y + 3 z + 3 y - z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

$- 9 - 2 y + 3 z + 3 y - z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 2 y$ and $3 y$ .

$- 9 + y + 3 z - z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

Step 2.2.1.2.2

Subtract $z$ from $3 z$ .

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$2 x - 5 y + 5 z = 17$

Step 2.3

Replace all occurrences of $x$ in $2 x - 5 y + 5 z = 17$ with $9 + 2 y - 3 z$ .

$2 (9 + 2 y - 3 z) - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (9 + 2 y - 3 z) - 5 y + 5 z$ .

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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.

$2 \cdot 9 + 2 (2 y) + 2 (- 3 z) - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $9$ .

$18 + 2 (2 y) + 2 (- 3 z) - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

Step 2.4.1.1.2.2

Multiply $2$ by $2$ .

$18 + 4 y + 2 (- 3 z) - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

Step 2.4.1.1.2.3

Multiply $- 3$ by $2$ .

$18 + 4 y - 6 z - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$18 + 4 y - 6 z - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$18 + 4 y - 6 z - 5 y + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $5 y$ from $4 y$ .

$18 - y - 6 z + 5 z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

Step 2.4.1.2.2

Add $- 6 z$ and $5 z$ .

$18 - y - z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$18 - y - z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$18 - y - z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$18 - y - z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

$18 - y - z = 17$

$- 9 + y + 2 z = - 6$

$x = 9 + 2 y - 3 z$

Step 3

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

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Step 3.1

Add $9$ to both sides of the equation.

$y + 2 z = - 6 + 9$

$18 - y - z = 17$

$x = 9 + 2 y - 3 z$

Step 3.2

Subtract $2 z$ from both sides of the equation.

$y = - 6 + 9 - 2 z$

$18 - y - z = 17$

$x = 9 + 2 y - 3 z$

Step 3.3

Add $- 6$ and $9$ .

$y = 3 - 2 z$

$18 - y - z = 17$

$x = 9 + 2 y - 3 z$

$y = 3 - 2 z$

$18 - y - z = 17$

$x = 9 + 2 y - 3 z$

Step 4

Replace all occurrences of $y$ with $3 - 2 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $18 - y - z = 17$ with $3 - 2 z$ .

$18 - (3 - 2 z) - z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $18 - (3 - 2 z) - z$ .

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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.

$18 - 1 \cdot 3 - (- 2 z) - z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

Step 4.2.1.1.2

Multiply $- 1$ by $3$ .

$18 - 3 - (- 2 z) - z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

Step 4.2.1.1.3

Multiply $- 2$ by $- 1$ .

$18 - 3 + 2 z - z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

$18 - 3 + 2 z - z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $3$ from $18$ .

$15 + 2 z - z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

Step 4.2.1.2.2

Subtract $z$ from $2 z$ .

$15 + z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

$15 + z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

$15 + z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

$15 + z = 17$

$y = 3 - 2 z$

$x = 9 + 2 y - 3 z$

Step 4.3

Replace all occurrences of $y$ in $x = 9 + 2 y - 3 z$ with $3 - 2 z$ .

$x = 9 + 2 (3 - 2 z) - 3 z$

$15 + z = 17$

$y = 3 - 2 z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $9 + 2 (3 - 2 z) - 3 z$ .

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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.

$x = 9 + 2 \cdot 3 + 2 (- 2 z) - 3 z$

$15 + z = 17$

$y = 3 - 2 z$

Step 4.4.1.1.2

Multiply $2$ by $3$ .

$x = 9 + 6 + 2 (- 2 z) - 3 z$

$15 + z = 17$

$y = 3 - 2 z$

Step 4.4.1.1.3

Multiply $- 2$ by $2$ .

$x = 9 + 6 - 4 z - 3 z$

$15 + z = 17$

$y = 3 - 2 z$

$x = 9 + 6 - 4 z - 3 z$

$15 + z = 17$

$y = 3 - 2 z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Add $9$ and $6$ .

$x = 15 - 4 z - 3 z$

$15 + z = 17$

$y = 3 - 2 z$

Step 4.4.1.2.2

Subtract $3 z$ from $- 4 z$ .

$x = 15 - 7 z$

$15 + z = 17$

$y = 3 - 2 z$

$x = 15 - 7 z$

$15 + z = 17$

$y = 3 - 2 z$

$x = 15 - 7 z$

$15 + z = 17$

$y = 3 - 2 z$

$x = 15 - 7 z$

$15 + z = 17$

$y = 3 - 2 z$

$x = 15 - 7 z$

$15 + z = 17$

$y = 3 - 2 z$

Step 5

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

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Step 5.1

Subtract $15$ from both sides of the equation.

$z = 17 - 15$

$x = 15 - 7 z$

$y = 3 - 2 z$

Step 5.2

Subtract $15$ from $17$ .

$z = 2$

$x = 15 - 7 z$

$y = 3 - 2 z$

$z = 2$

$x = 15 - 7 z$

$y = 3 - 2 z$

Step 6

Replace all occurrences of $z$ with $2$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = 15 - 7 z$ with $2$ .

$x = 15 - 7 \cdot 2$

$z = 2$

$y = 3 - 2 z$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $15 - 7 \cdot 2$ .

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Step 6.2.1.1

Multiply $- 7$ by $2$ .

$x = 15 - 14$

$z = 2$

$y = 3 - 2 z$

Step 6.2.1.2

Subtract $14$ from $15$ .

$x = 1$

$z = 2$

$y = 3 - 2 z$

$x = 1$

$z = 2$

$y = 3 - 2 z$

$x = 1$

$z = 2$

$y = 3 - 2 z$

Step 6.3

Replace all occurrences of $z$ in $y = 3 - 2 z$ with $2$ .

$y = 3 - 2 \cdot 2$

$x = 1$

$z = 2$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $3 - 2 \cdot 2$ .

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Step 6.4.1.1

Multiply $- 2$ by $2$ .

$y = 3 - 4$

$x = 1$

$z = 2$

Step 6.4.1.2

Subtract $4$ from $3$ .

$y = - 1$

$x = 1$

$z = 2$

$y = - 1$

$x = 1$

$z = 2$

$y = - 1$

$x = 1$

$z = 2$

$y = - 1$

$x = 1$

$z = 2$

Step 7

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

$(1, - 1, 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, - 1, 2)$

Equation Form:

$x = 1, y = - 1, z = 2$