Algebra Examples

$x - y - 2 z = - 2$ $2 x - y + z = - 1$ $- 7 x + 5 y + 4 z = 8$

Step 1

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

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Step 1.1

Add $y$ to both sides of the equation.

$x - 2 z = - 2 + y$

$2 x - y + z = - 1$

$- 7 x + 5 y + 4 z = 8$

Step 1.2

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

$x = - 2 + y + 2 z$

$2 x - y + z = - 1$

$- 7 x + 5 y + 4 z = 8$

$x = - 2 + y + 2 z$

$2 x - y + z = - 1$

$- 7 x + 5 y + 4 z = 8$

Step 2

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

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Step 2.1

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

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

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $- 2$ .

$- 4 + 2 y + 2 (2 z) - y + z = - 1$

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

Step 2.2.1.1.2.2

Multiply $2$ by $2$ .

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

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

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

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

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

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $y$ from $2 y$ .

$- 4 + y + 4 z + z = - 1$

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

Step 2.2.1.2.2

Add $4 z$ and $z$ .

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$- 7 x + 5 y + 4 z = 8$

Step 2.3

Replace all occurrences of $x$ in $- 7 x + 5 y + 4 z = 8$ with $- 2 + y + 2 z$ .

$- 7 (- 2 + y + 2 z) + 5 y + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 7 (- 2 + y + 2 z) + 5 y + 4 z$ .

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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.

$- 7 \cdot - 2 - 7 y - 7 (2 z) + 5 y + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 7$ by $- 2$ .

$14 - 7 y - 7 (2 z) + 5 y + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

Step 2.4.1.1.2.2

Multiply $2$ by $- 7$ .

$14 - 7 y - 14 z + 5 y + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$14 - 7 y - 14 z + 5 y + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$14 - 7 y - 14 z + 5 y + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 7 y$ and $5 y$ .

$14 - 2 y - 14 z + 4 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

Step 2.4.1.2.2

Add $- 14 z$ and $4 z$ .

$14 - 2 y - 10 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$14 - 2 y - 10 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$14 - 2 y - 10 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$14 - 2 y - 10 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

$14 - 2 y - 10 z = 8$

$- 4 + y + 5 z = - 1$

$x = - 2 + y + 2 z$

Step 3

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

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Step 3.1

Add $4$ to both sides of the equation.

$y + 5 z = - 1 + 4$

$14 - 2 y - 10 z = 8$

$x = - 2 + y + 2 z$

Step 3.2

Subtract $5 z$ from both sides of the equation.

$y = - 1 + 4 - 5 z$

$14 - 2 y - 10 z = 8$

$x = - 2 + y + 2 z$

Step 3.3

Add $- 1$ and $4$ .

$y = 3 - 5 z$

$14 - 2 y - 10 z = 8$

$x = - 2 + y + 2 z$

$y = 3 - 5 z$

$14 - 2 y - 10 z = 8$

$x = - 2 + y + 2 z$

Step 4

Replace all occurrences of $y$ with $3 - 5 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $14 - 2 y - 10 z = 8$ with $3 - 5 z$ .

$14 - 2 (3 - 5 z) - 10 z = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $14 - 2 (3 - 5 z) - 10 z$ .

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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.

$14 - 2 \cdot 3 - 2 (- 5 z) - 10 z = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

Step 4.2.1.1.2

Multiply $- 2$ by $3$ .

$14 - 6 - 2 (- 5 z) - 10 z = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

Step 4.2.1.1.3

Multiply $- 5$ by $- 2$ .

$14 - 6 + 10 z - 10 z = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

$14 - 6 + 10 z - 10 z = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $14 - 6 + 10 z - 10 z$ .

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Step 4.2.1.2.1.1

Subtract $10 z$ from $10 z$ .

$14 - 6 + 0 = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

Step 4.2.1.2.1.2

Add $14 - 6$ and $0$ .

$14 - 6 = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

$14 - 6 = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

Step 4.2.1.2.2

Subtract $6$ from $14$ .

$8 = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

$8 = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

$8 = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

$8 = 8$

$y = 3 - 5 z$

$x = - 2 + y + 2 z$

Step 4.3

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

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

$8 = 8$

$y = 3 - 5 z$

Step 4.4

Simplify $x = - 2 + 3 - 5 z + 2 z$ .

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Remove parentheses.

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

$8 = 8$

$y = 3 - 5 z$

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

$8 = 8$

$y = 3 - 5 z$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $- 2 + 3 - 5 z + 2 z$ .

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Step 4.4.2.1.1

Add $- 2$ and $3$ .

$x = 1 - 5 z + 2 z$

$8 = 8$

$y = 3 - 5 z$

Step 4.4.2.1.2

Add $- 5 z$ and $2 z$ .

$x = 1 - 3 z$

$8 = 8$

$y = 3 - 5 z$

$x = 1 - 3 z$

$8 = 8$

$y = 3 - 5 z$

$x = 1 - 3 z$

$8 = 8$

$y = 3 - 5 z$

$x = 1 - 3 z$

$8 = 8$

$y = 3 - 5 z$

$x = 1 - 3 z$

$8 = 8$

$y = 3 - 5 z$

Step 5

Remove any equations from the system that are always true.

$x = 1 - 3 z$

$y = 3 - 5 z$