Algebra Examples

$- 3 x - y + 2 z = 1$ $- x + y - z = 2$ $- x - 3 y + 4 z = - 3$

Step 1

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

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Step 1.1

Add $x$ to both sides of the equation.

$y - z = 2 + x$

$- 3 x - y + 2 z = 1$

$- x - 3 y + 4 z = - 3$

Step 1.2

Add $z$ to both sides of the equation.

$y = 2 + x + z$

$- 3 x - y + 2 z = 1$

$- x - 3 y + 4 z = - 3$

$y = 2 + x + z$

$- 3 x - y + 2 z = 1$

$- x - 3 y + 4 z = - 3$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $- 3 x - y + 2 z = 1$ with $2 + x + z$ .

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

$y = 2 + x + z$

$- x - 3 y + 4 z = - 3$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$- 3 x - 1 \cdot 2 - x - z + 2 z = 1$

$y = 2 + x + z$

$- x - 3 y + 4 z = - 3$

Step 2.2.1.1.2

Multiply $- 1$ by $2$ .

$- 3 x - 2 - x - z + 2 z = 1$

$y = 2 + x + z$

$- x - 3 y + 4 z = - 3$

$- 3 x - 2 - x - z + 2 z = 1$

$y = 2 + x + z$

$- x - 3 y + 4 z = - 3$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $x$ from $- 3 x$ .

$- 4 x - 2 - z + 2 z = 1$

$y = 2 + x + z$

$- x - 3 y + 4 z = - 3$

Step 2.2.1.2.2

Add $- z$ and $2 z$ .

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$- x - 3 y + 4 z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$- x - 3 y + 4 z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$- x - 3 y + 4 z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$- x - 3 y + 4 z = - 3$

Step 2.3

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

$- x - 3 (2 + x + z) + 4 z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- x - 3 (2 + x + z) + 4 z$ .

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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.

$- x - 3 \cdot 2 - 3 x - 3 z + 4 z = - 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Step 2.4.1.1.2

Multiply $- 3$ by $2$ .

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $3 x$ from $- x$ .

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Step 2.4.1.2.2

Add $- 3 z$ and $4 z$ .

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Step 3

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

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Step 3.1

Add $4 x$ to both sides of the equation.

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

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Step 3.2

Add $6$ to both sides of the equation.

$z = - 3 + 4 x + 6$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Step 3.3

Add $- 3$ and $6$ .

$z = 4 x + 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

$z = 4 x + 3$

$- 4 x - 2 + z = 1$

$y = 2 + x + z$

Step 4

Replace all occurrences of $z$ with $4 x + 3$ in each equation.

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Step 4.1

Replace all occurrences of $z$ in $- 4 x - 2 + z = 1$ with $4 x + 3$ .

$- 4 x - 2 + 4 x + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 4 x - 2 + 4 x + 3$ .

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Step 4.2.1.1

Remove parentheses.

$- 4 x - 2 + 4 x + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

Step 4.2.1.2

Combine the opposite terms in $- 4 x - 2 + 4 x + 3$ .

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Step 4.2.1.2.1

Add $- 4 x$ and $4 x$ .

$0 - 2 + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

Step 4.2.1.2.2

Subtract $2$ from $0$ .

$- 2 + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

$- 2 + 3 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

Step 4.2.1.3

Add $- 2$ and $3$ .

$1 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

$1 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

$1 = 1$

$z = 4 x + 3$

$y = 2 + x + z$

Step 4.3

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

$y = 2 + x + 4 x + 3$

$1 = 1$

$z = 4 x + 3$

Step 4.4

Simplify $y = 2 + x + 4 x + 3$ .

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Remove parentheses.

$y = 2 + x + 4 x + 3$

$1 = 1$

$z = 4 x + 3$

$y = 2 + x + 4 x + 3$

$1 = 1$

$z = 4 x + 3$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $2 + x + 4 x + 3$ .

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Step 4.4.2.1.1

Add $2$ and $3$ .

$y = x + 4 x + 5$

$1 = 1$

$z = 4 x + 3$

Step 4.4.2.1.2

Add $x$ and $4 x$ .

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

$y = 5 x + 5$

$1 = 1$

$z = 4 x + 3$

Step 5

Remove any equations from the system that are always true.

$y = 5 x + 5$

$z = 4 x + 3$