Algebra Examples

$3 x + 2 y - z = - 5$ $- x - y + z = 1$ $5 x + 4 y - 3 z = - 7$

Step 1

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

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Step 1.1

Add $x$ to both sides of the equation.

$- y + z = 1 + x$

$3 x + 2 y - z = - 5$

$5 x + 4 y - 3 z = - 7$

Step 1.2

Add $y$ to both sides of the equation.

$z = 1 + x + y$

$3 x + 2 y - z = - 5$

$5 x + 4 y - 3 z = - 7$

$z = 1 + x + y$

$3 x + 2 y - z = - 5$

$5 x + 4 y - 3 z = - 7$

Step 2

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

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Step 2.1

Replace all occurrences of $z$ in $3 x + 2 y - z = - 5$ with $1 + x + y$ .

$3 x + 2 y - (1 + x + y) = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 z = - 7$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 x + 2 y - (1 + x + y)$ .

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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.

$3 x + 2 y - 1 \cdot 1 - x - y = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 z = - 7$

Step 2.2.1.1.2

Multiply $- 1$ by $1$ .

$3 x + 2 y - 1 - x - y = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 z = - 7$

$3 x + 2 y - 1 - x - y = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 z = - 7$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $x$ from $3 x$ .

$2 x + 2 y - 1 - y = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 z = - 7$

Step 2.2.1.2.2

Subtract $y$ from $2 y$ .

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 z = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 z = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 z = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 z = - 7$

Step 2.3

Replace all occurrences of $z$ in $5 x + 4 y - 3 z = - 7$ with $1 + x + y$ .

$5 x + 4 y - 3 (1 + x + y) = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $5 x + 4 y - 3 (1 + x + y)$ .

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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.

$5 x + 4 y - 3 \cdot 1 - 3 x - 3 y = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

Step 2.4.1.1.2

Multiply $- 3$ by $1$ .

$5 x + 4 y - 3 - 3 x - 3 y = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$5 x + 4 y - 3 - 3 x - 3 y = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $3 x$ from $5 x$ .

$2 x + 4 y - 3 - 3 y = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

Step 2.4.1.2.2

Subtract $3 y$ from $4 y$ .

$2 x + y - 3 = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$2 x + y - 3 = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$2 x + y - 3 = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$2 x + y - 3 = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$2 x + y - 3 = - 7$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

Step 3

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

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Step 3.1

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

$y - 3 = - 7 - 2 x$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

Step 3.2

Add $3$ to both sides of the equation.

$y = - 7 - 2 x + 3$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

Step 3.3

Add $- 7$ and $3$ .

$y = - 2 x - 4$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

$y = - 2 x - 4$

$2 x + y - 1 = - 5$

$z = 1 + x + y$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $2 x + y - 1 = - 5$ with $- 2 x - 4$ .

$2 x - 2 x - 4 - 1 = - 5$

$y = - 2 x - 4$

$z = 1 + x + y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $2 x - 2 x - 4 - 1$ .

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Step 4.2.1.1

Remove parentheses.

$2 x - 2 x - 4 - 1 = - 5$

$y = - 2 x - 4$

$z = 1 + x + y$

Step 4.2.1.2

Subtract $2 x$ from $2 x$ .

$0 - 4 - 1 = - 5$

$y = - 2 x - 4$

$z = 1 + x + y$

Step 4.2.1.3

Simplify by subtracting numbers.

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Step 4.2.1.3.1

Subtract $4$ from $0$ .

$- 4 - 1 = - 5$

$y = - 2 x - 4$

$z = 1 + x + y$

Step 4.2.1.3.2

Subtract $1$ from $- 4$ .

$- 5 = - 5$

$y = - 2 x - 4$

$z = 1 + x + y$

$- 5 = - 5$

$y = - 2 x - 4$

$z = 1 + x + y$

$- 5 = - 5$

$y = - 2 x - 4$

$z = 1 + x + y$

$- 5 = - 5$

$y = - 2 x - 4$

$z = 1 + x + y$

Step 4.3

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

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

$- 5 = - 5$

$y = - 2 x - 4$

Step 4.4

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

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Remove parentheses.

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

$- 5 = - 5$

$y = - 2 x - 4$

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

$- 5 = - 5$

$y = - 2 x - 4$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $1 + x - 2 x - 4$ .

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Step 4.4.2.1.1

Subtract $4$ from $1$ .

$z = x - 2 x - 3$

$- 5 = - 5$

$y = - 2 x - 4$

Step 4.4.2.1.2

Subtract $2 x$ from $x$ .

$z = - x - 3$

$- 5 = - 5$

$y = - 2 x - 4$

$z = - x - 3$

$- 5 = - 5$

$y = - 2 x - 4$

$z = - x - 3$

$- 5 = - 5$

$y = - 2 x - 4$

$z = - x - 3$

$- 5 = - 5$

$y = - 2 x - 4$

$z = - x - 3$

$- 5 = - 5$

$y = - 2 x - 4$

Step 5

Remove any equations from the system that are always true.

$z = - x - 3$

$y = - 2 x - 4$