Algebra Examples

$- x + 3 y - 2 z = 1$ $2 x - 6 y + 4 z = - 2$ $7 x - 14 y + 21 z = 0$

Step 1

Solve for $x$ in $- x + 3 y - 2 z = 1$ .

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Step 1.1

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

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Step 1.1.1

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

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.1.2

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

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2

Divide each term in $- x = 1 - 3 y + 2 z$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- x = 1 - 3 y + 2 z$ by $- 1$ .

$\frac{- x}{- 1} = \frac{1}{- 1} + \frac{- 3 y}{- 1} + \frac{2 z}{- 1}$

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{1}{- 1} + \frac{- 3 y}{- 1} + \frac{2 z}{- 1}$

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2.2.2

Divide $x$ by $1$ .

$x = \frac{1}{- 1} + \frac{- 3 y}{- 1} + \frac{2 z}{- 1}$

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

$x = \frac{1}{- 1} + \frac{- 3 y}{- 1} + \frac{2 z}{- 1}$

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $1$ by $- 1$ .

$x = - 1 + \frac{- 3 y}{- 1} + \frac{2 z}{- 1}$

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2.3.1.2

Move the negative one from the denominator of $\frac{- 3 y}{- 1}$ .

$x = - 1 - 1 \cdot (- 3 y) + \frac{2 z}{- 1}$

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2.3.1.3

Rewrite $- 1 \cdot (- 3 y)$ as $- (- 3 y)$ .

$x = - 1 - (- 3 y) + \frac{2 z}{- 1}$

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2.3.1.4

Multiply $- 3$ by $- 1$ .

$x = - 1 + 3 y + \frac{2 z}{- 1}$

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2.3.1.5

Move the negative one from the denominator of $\frac{2 z}{- 1}$ .

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2.3.1.6

Rewrite $- 1 \cdot (2 z)$ as $- (2 z)$ .

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 1.2.3.1.7

Multiply $2$ by $- 1$ .

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

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

$2 x - 6 y + 4 z = - 2$

$7 x - 14 y + 21 z = 0$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $2 x - 6 y + 4 z = - 2$ with $- 1 + 3 y - 2 z$ .

$2 (- 1 + 3 y - 2 z) - 6 y + 4 z = - 2$

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

$7 x - 14 y + 21 z = 0$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- 1 + 3 y - 2 z) - 6 y + 4 z$ .

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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.

$2 \cdot - 1 + 2 (3 y) + 2 (- 2 z) - 6 y + 4 z = - 2$

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

$7 x - 14 y + 21 z = 0$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $- 1$ .

$- 2 + 2 (3 y) + 2 (- 2 z) - 6 y + 4 z = - 2$

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

$7 x - 14 y + 21 z = 0$

Step 2.2.1.1.2.2

Multiply $3$ by $2$ .

$- 2 + 6 y + 2 (- 2 z) - 6 y + 4 z = - 2$

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

$7 x - 14 y + 21 z = 0$

Step 2.2.1.1.2.3

Multiply $- 2$ by $2$ .

$- 2 + 6 y - 4 z - 6 y + 4 z = - 2$

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

$7 x - 14 y + 21 z = 0$

$- 2 + 6 y - 4 z - 6 y + 4 z = - 2$

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

$7 x - 14 y + 21 z = 0$

$- 2 + 6 y - 4 z - 6 y + 4 z = - 2$

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

$7 x - 14 y + 21 z = 0$

Step 2.2.1.2

Combine the opposite terms in $- 2 + 6 y - 4 z - 6 y + 4 z$ .

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Step 2.2.1.2.1

Subtract $6 y$ from $6 y$ .

$- 2 - 4 z + 0 + 4 z = - 2$

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

$7 x - 14 y + 21 z = 0$

Step 2.2.1.2.2

Add $- 2 - 4 z$ and $0$ .

$- 2 - 4 z + 4 z = - 2$

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

$7 x - 14 y + 21 z = 0$

Step 2.2.1.2.3

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

$- 2 + 0 = - 2$

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

$7 x - 14 y + 21 z = 0$

Step 2.2.1.2.4

Add $- 2$ and $0$ .

$- 2 = - 2$

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

$7 x - 14 y + 21 z = 0$

$- 2 = - 2$

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

$7 x - 14 y + 21 z = 0$

$- 2 = - 2$

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

$7 x - 14 y + 21 z = 0$

$- 2 = - 2$

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

$7 x - 14 y + 21 z = 0$

Step 2.3

Replace all occurrences of $x$ in $7 x - 14 y + 21 z = 0$ with $- 1 + 3 y - 2 z$ .

$7 (- 1 + 3 y - 2 z) - 14 y + 21 z = 0$

$- 2 = - 2$

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

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $7 (- 1 + 3 y - 2 z) - 14 y + 21 z$ .

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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.

$7 \cdot - 1 + 7 (3 y) + 7 (- 2 z) - 14 y + 21 z = 0$

$- 2 = - 2$

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

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $7$ by $- 1$ .

$- 7 + 7 (3 y) + 7 (- 2 z) - 14 y + 21 z = 0$

$- 2 = - 2$

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

Step 2.4.1.1.2.2

Multiply $3$ by $7$ .

$- 7 + 21 y + 7 (- 2 z) - 14 y + 21 z = 0$

$- 2 = - 2$

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

Step 2.4.1.1.2.3

Multiply $- 2$ by $7$ .

$- 7 + 21 y - 14 z - 14 y + 21 z = 0$

$- 2 = - 2$

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

$- 7 + 21 y - 14 z - 14 y + 21 z = 0$

$- 2 = - 2$

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

$- 7 + 21 y - 14 z - 14 y + 21 z = 0$

$- 2 = - 2$

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

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $14 y$ from $21 y$ .

$- 7 + 7 y - 14 z + 21 z = 0$

$- 2 = - 2$

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

Step 2.4.1.2.2

Add $- 14 z$ and $21 z$ .

$- 7 + 7 y + 7 z = 0$

$- 2 = - 2$

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

$- 7 + 7 y + 7 z = 0$

$- 2 = - 2$

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

$- 7 + 7 y + 7 z = 0$

$- 2 = - 2$

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

$- 7 + 7 y + 7 z = 0$

$- 2 = - 2$

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

$- 7 + 7 y + 7 z = 0$

$- 2 = - 2$

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

Step 3

Remove any equations from the system that are always true.

$- 7 + 7 y + 7 z = 0$

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