Algebra Examples

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

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.

$x - 2 z = 3 - y$

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

$x + 2 y - 2 z = 6$

Step 1.2

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

$x = 3 - y + 2 z$

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

$x + 2 y - 2 z = 6$

$x = 3 - y + 2 z$

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

$x + 2 y - 2 z = 6$

Step 2

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

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Step 2.1

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

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

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $3$ .

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

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

Step 2.2.1.1.2.2

Multiply $- - y$ .

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Step 2.2.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$- 3 + 1 y - (2 z) - y + 3 z = - 2$

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

Step 2.2.1.1.2.2.2

Multiply $y$ by $1$ .

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

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

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

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

Step 2.2.1.1.2.3

Multiply $2$ by $- 1$ .

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

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

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

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

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

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Combine the opposite terms in $- 3 + y - 2 z - y + 3 z$ .

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Step 2.2.1.2.1.1

Subtract $y$ from $y$ .

$- 3 - 2 z + 0 + 3 z = - 2$

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

Step 2.2.1.2.1.2

Add $- 3 - 2 z$ and $0$ .

$- 3 - 2 z + 3 z = - 2$

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

$- 3 - 2 z + 3 z = - 2$

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

Step 2.2.1.2.2

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

$- 3 + z = - 2$

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

$x + 2 y - 2 z = 6$

Step 2.3

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

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

$- 3 + z = - 2$

$x = 3 - y + 2 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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Step 2.4.1.1

Combine the opposite terms in $(3 - y + 2 z) + 2 y - 2 z$ .

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Step 2.4.1.1.1

Subtract $2 z$ from $2 z$ .

$(3 - y) + 2 y + 0 = 6$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

Step 2.4.1.1.2

Add $(3 - y) + 2 y$ and $0$ .

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

$- 3 + z = - 2$

$x = 3 - y + 2 z$

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

$- 3 + z = - 2$

$x = 3 - y + 2 z$

Step 2.4.1.2

Add $- y$ and $2 y$ .

$3 + y = 6$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

$3 + y = 6$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

$3 + y = 6$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

$3 + y = 6$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

Step 3

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

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Step 3.1

Subtract $3$ from both sides of the equation.

$y = 6 - 3$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

Step 3.2

Subtract $3$ from $6$ .

$y = 3$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

$y = 3$

$- 3 + z = - 2$

$x = 3 - y + 2 z$

Step 4

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

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Step 4.1

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

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Step 4.1.1

Add $3$ to both sides of the equation.

$z = - 2 + 3$

$y = 3$

$x = 3 - y + 2 z$

Step 4.1.2

Add $- 2$ and $3$ .

$z = 1$

$y = 3$

$x = 3 - y + 2 z$

$z = 1$

$y = 3$

$x = 3 - y + 2 z$

Step 4.2

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

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

$z = 1$

$y = 3$

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Multiply $- 1$ by $3$ .

$x = 3 - 3 + 2 z$

$z = 1$

$y = 3$

Step 4.3.1.2

Subtract $3$ from $3$ .

$x = 0 + 2 z$

$z = 1$

$y = 3$

Step 4.3.1.3

Add $0$ and $2 z$ .

$x = 2 z$

$z = 1$

$y = 3$

$x = 2 z$

$z = 1$

$y = 3$

$x = 2 z$

$z = 1$

$y = 3$

$x = 2 z$

$z = 1$

$y = 3$

Step 5

Replace all occurrences of $z$ with $1$ in each equation.

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Step 5.1

Replace all occurrences of $z$ in $x = 2 z$ with $1$ .

$x = 2 (1)$

$z = 1$

$y = 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $2$ by $1$ .

$x = 2$

$z = 1$

$y = 3$

$x = 2$

$z = 1$

$y = 3$

$x = 2$

$z = 1$

$y = 3$

Step 6

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

$(2, 3, 1)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(2, 3, 1)$

Equation Form:

$x = 2, y = 3, z = 1$