Solve the System of Equations 3x-y+2z=4 6x-2y+4z=-8 2x-y+3z=10

$3 x - y + 2 z = 4$ $6 x - 2 y + 4 z = - 8$ $2 x - y + 3 z = 10$

Solve for $x$ in $3 x - y + 2 z = 4$ .

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

Add $y$ to both sides of the equation.

$3 x + 2 z = 4 + y$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

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

$3 x = 4 + y - 2 z$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

$3 x = 4 + y - 2 z$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

Divide each term in $3 x = 4 + y - 2 z$ by $3$ and simplify.

Divide each term in $3 x = 4 + y - 2 z$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3} + \frac{y}{3} + \frac{- 2 z}{3}$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3} + \frac{y}{3} + \frac{- 2 z}{3}$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

Divide $x$ by $1$ .

$x = \frac{4}{3} + \frac{y}{3} + \frac{- 2 z}{3}$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

$x = \frac{4}{3} + \frac{y}{3} + \frac{- 2 z}{3}$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

$x = \frac{4}{3} + \frac{y}{3} + \frac{- 2 z}{3}$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$6 x - 2 y + 4 z = - 8$

$2 x - y + 3 z = 10$

Replace all occurrences of $x$ with $\frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$ in each equation.

Replace all occurrences of $x$ in $6 x - 2 y + 4 z = - 8$ with $\frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$ .

$6 (\frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Simplify the left side.

Simplify $6 (\frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}) - 2 y + 4 z$ .

Simplify each term.

Apply the distributive property.

$6 (\frac{4}{3}) + 6 (\frac{y}{3}) + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Simplify.

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$3 (2) (\frac{4}{3}) + 6 (\frac{y}{3}) + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Cancel the common factor.

$3 \cdot (2 (\frac{4}{3})) + 6 (\frac{y}{3}) + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Rewrite the expression.

$2 \cdot 4 + 6 (\frac{y}{3}) + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

$2 \cdot 4 + 6 (\frac{y}{3}) + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Multiply $2$ by $4$ .

$8 + 6 (\frac{y}{3}) + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$8 + 3 (2) (\frac{y}{3}) + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Cancel the common factor.

$8 + 3 \cdot (2 (\frac{y}{3})) + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Rewrite the expression.

$8 + 2 y + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

$8 + 2 y + 6 (- \frac{2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{2 z}{3}$ into the numerator.

$8 + 2 y + 6 (\frac{- 2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Factor $3$ out of $6$ .

$8 + 2 y + 3 (2) (\frac{- 2 z}{3}) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Cancel the common factor.

$8 + 2 y + 3 \cdot (2 (\frac{- 2 z}{3})) - 2 y + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Rewrite the expression.

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

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

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

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Multiply $- 2$ by $2$ .

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

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

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

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

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

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Combine the opposite terms in $8 + 2 y - 4 z - 2 y + 4 z$ .

Subtract $2 y$ from $2 y$ .

$8 - 4 z + 0 + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Add $8 - 4 z$ and $0$ .

$8 - 4 z + 4 z = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

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

$8 + 0 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Add $8$ and $0$ .

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$2 x - y + 3 z = 10$

Replace all occurrences of $x$ in $2 x - y + 3 z = 10$ with $\frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$ .

$2 (\frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}) - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Simplify the left side.

Simplify $2 (\frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}) - y + 3 z$ .

Simplify each term.

Apply the distributive property.

$2 (\frac{4}{3}) + 2 (\frac{y}{3}) + 2 (- \frac{2 z}{3}) - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Simplify.

Multiply $2 (\frac{4}{3})$ .

Combine $2$ and $\frac{4}{3}$ .

$\frac{2 \cdot 4}{3} + 2 (\frac{y}{3}) + 2 (- \frac{2 z}{3}) - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Multiply $2$ by $4$ .

$\frac{8}{3} + 2 (\frac{y}{3}) + 2 (- \frac{2 z}{3}) - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$\frac{8}{3} + 2 (\frac{y}{3}) + 2 (- \frac{2 z}{3}) - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Combine $2$ and $\frac{y}{3}$ .

$\frac{8}{3} + \frac{2 y}{3} + 2 (- \frac{2 z}{3}) - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Multiply $2 (- \frac{2 z}{3})$ .

Multiply $- 1$ by $2$ .

$\frac{8}{3} + \frac{2 y}{3} - 2 \frac{2 z}{3} - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Combine $- 2$ and $\frac{2 z}{3}$ .

$\frac{8}{3} + \frac{2 y}{3} + \frac{- 2 (2 z)}{3} - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Multiply $2$ by $- 2$ .

$\frac{8}{3} + \frac{2 y}{3} + \frac{- 4 z}{3} - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$\frac{8}{3} + \frac{2 y}{3} + \frac{- 4 z}{3} - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$\frac{8}{3} + \frac{2 y}{3} + \frac{- 4 z}{3} - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Move the negative in front of the fraction.

$\frac{8}{3} + \frac{2 y}{3} - \frac{4 z}{3} - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$\frac{8}{3} + \frac{2 y}{3} - \frac{4 z}{3} - y + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

To write $- y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} - \frac{4 z}{3} + \frac{2 y}{3} - y \cdot \frac{3}{3} + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Combine $- y$ and $\frac{3}{3}$ .

$\frac{8}{3} - \frac{4 z}{3} + \frac{2 y}{3} + \frac{- y \cdot 3}{3} + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Combine the numerators over the common denominator.

$\frac{8}{3} - \frac{4 z}{3} + \frac{2 y - y \cdot 3}{3} + 3 z = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Combine the numerators over the common denominator.

$3 z + \frac{8 - 4 z + 2 y - y \cdot 3}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Multiply $3$ by $- 1$ .

$3 z + \frac{8 - 4 z + 2 y - 3 y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Subtract $3 y$ from $2 y$ .

$3 z + \frac{8 - 4 z - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

To write $3 z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 z \cdot \frac{3}{3} + \frac{8 - 4 z - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Simplify terms.

Combine $3 z$ and $\frac{3}{3}$ .

$\frac{3 z \cdot 3}{3} + \frac{8 - 4 z - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Combine the numerators over the common denominator.

$\frac{3 z \cdot 3 + 8 - 4 z - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$\frac{3 z \cdot 3 + 8 - 4 z - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Simplify the numerator.

Multiply $3$ by $3$ .

$\frac{9 z + 8 - 4 z - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Subtract $4 z$ from $9 z$ .

$\frac{5 z + 8 - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$\frac{5 z + 8 - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$\frac{5 z + 8 - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$\frac{5 z + 8 - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

$\frac{5 z + 8 - y}{3} = 10$

$8 = - 8$

$x = \frac{4}{3} + \frac{y}{3} - \frac{2 z}{3}$

Since $8 = - 8$ is not true, there is no solution.

No solution

Solve the System of Equations 3x-y+2z=4 6x-2y+4z=-8 2x-y+3z=10

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