Solve the System of Equations -x+3y-2z=1 2x-6y+4z=-2 7x-14y+21z=0

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

Step 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Divide $x$ by $1$ .

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

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

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

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

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

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

Simplify the right side.

Simplify each term.

Divide $1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

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

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

Multiply $- 3$ by $- 1$ .

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

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

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

Simplify.

Multiply $2$ by $- 1$ .

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

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

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

Multiply $3$ by $2$ .

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

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

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

Multiply $- 2$ by $2$ .

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

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

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

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

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

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

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

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

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

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

Subtract $6 y$ from $6 y$ .

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

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

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

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

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

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

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

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

$- 2 + 0 = - 2$

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

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

Add $- 2$ and $0$ .

$- 2 = - 2$

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

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

$- 2 = - 2$

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

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

$- 2 = - 2$

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

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

$- 2 = - 2$

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

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

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

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

$- 2 = - 2$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$- 2 = - 2$

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

Simplify.

Multiply $7$ by $- 1$ .

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

$- 2 = - 2$

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

Multiply $3$ by $7$ .

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

$- 2 = - 2$

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

Multiply $- 2$ by $7$ .

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

$- 2 = - 2$

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

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

$- 2 = - 2$

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

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

$- 2 = - 2$

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

Simplify by adding terms.

Subtract $14 y$ from $21 y$ .

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

$- 2 = - 2$

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

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.

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

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

Solve the System of Equations -x+3y-2z=1 2x-6y+4z=-2 7x-14y+21z=0

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