Solve the System of Equations x+y+z=3 13x+2z=2 -x-5z=-5

$x + y + z = 3$ $13 x + 2 z = 2$ $- x - 5 z = - 5$

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

Subtract $y$ from both sides of the equation.

$x + z = 3 - y$

$13 x + 2 z = 2$

$- x - 5 z = - 5$

Subtract $z$ from both sides of the equation.

$x = 3 - y - z$

$13 x + 2 z = 2$

$- x - 5 z = - 5$

$x = 3 - y - z$

$13 x + 2 z = 2$

$- x - 5 z = - 5$

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

Replace all occurrences of $x$ in $13 x + 2 z = 2$ with $3 - y - z$ .

$13 (3 - y - z) + 2 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

Simplify the left side.

Simplify $13 (3 - y - z) + 2 z$ .

Simplify each term.

Apply the distributive property.

$13 \cdot 3 + 13 (- y) + 13 (- z) + 2 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

Simplify.

Multiply $13$ by $3$ .

$39 + 13 (- y) + 13 (- z) + 2 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

Multiply $- 1$ by $13$ .

$39 - 13 y + 13 (- z) + 2 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

Multiply $- 1$ by $13$ .

$39 - 13 y - 13 z + 2 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

$39 - 13 y - 13 z + 2 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

$39 - 13 y - 13 z + 2 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

Add $- 13 z$ and $2 z$ .

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

$- x - 5 z = - 5$

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

$- (3 - y - z) - 5 z = - 5$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

Simplify the left side.

Simplify $- (3 - y - z) - 5 z$ .

Simplify each term.

Apply the distributive property.

$- 1 \cdot 3 + y + z - 5 z = - 5$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

Simplify.

Multiply $- 1$ by $3$ .

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

Multiply $- - y$ .

Multiply $- 1$ by $- 1$ .

$- 3 + 1 y + z - 5 z = - 5$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

Multiply $y$ by $1$ .

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

Multiply $- - z$ .

Multiply $- 1$ by $- 1$ .

$- 3 + y + 1 z - 5 z = - 5$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

Multiply $z$ by $1$ .

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

Subtract $5 z$ from $z$ .

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

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

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

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

Add $3$ to both sides of the equation.

$y - 4 z = - 5 + 3$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

Add $4 z$ to both sides of the equation.

$y = - 5 + 3 + 4 z$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

Add $- 5$ and $3$ .

$y = - 2 + 4 z$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

$y = - 2 + 4 z$

$39 - 13 y - 11 z = 2$

$x = 3 - y - z$

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

Replace all occurrences of $y$ in $39 - 13 y - 11 z = 2$ with $- 2 + 4 z$ .

$39 - 13 (- 2 + 4 z) - 11 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

Simplify the left side.

Simplify $39 - 13 (- 2 + 4 z) - 11 z$ .

Simplify each term.

Apply the distributive property.

$39 - 13 \cdot - 2 - 13 (4 z) - 11 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

Multiply $- 13$ by $- 2$ .

$39 + 26 - 13 (4 z) - 11 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

Multiply $4$ by $- 13$ .

$39 + 26 - 52 z - 11 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

$39 + 26 - 52 z - 11 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

Simplify by adding terms.

Add $39$ and $26$ .

$65 - 52 z - 11 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

Subtract $11 z$ from $- 52 z$ .

$65 - 63 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

$x = 3 - y - z$

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

$x = 3 - (- 2 + 4 z) - z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

Simplify the right side.

Simplify $3 - (- 2 + 4 z) - z$ .

Simplify each term.

Apply the distributive property.

$x = 3 + 2 - (4 z) - z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

Multiply $- 1$ by $- 2$ .

$x = 3 + 2 - (4 z) - z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

Multiply $4$ by $- 1$ .

$x = 3 + 2 - 4 z - z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

$x = 3 + 2 - 4 z - z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

Simplify by adding terms.

Add $3$ and $2$ .

$x = 5 - 4 z - z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

Subtract $z$ from $- 4 z$ .

$x = 5 - 5 z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

$x = 5 - 5 z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

$x = 5 - 5 z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

$x = 5 - 5 z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

$x = 5 - 5 z$

$65 - 63 z = 2$

$y = - 2 + 4 z$

Solve for $z$ in $65 - 63 z = 2$ .

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

Subtract $65$ from both sides of the equation.

$- 63 z = 2 - 65$

$x = 5 - 5 z$

$y = - 2 + 4 z$

Subtract $65$ from $2$ .

$- 63 z = - 63$

$x = 5 - 5 z$

$y = - 2 + 4 z$

$- 63 z = - 63$

$x = 5 - 5 z$

$y = - 2 + 4 z$

Divide each term in $- 63 z = - 63$ by $- 63$ and simplify.

Divide each term in $- 63 z = - 63$ by $- 63$ .

$\frac{- 63 z}{- 63} = \frac{- 63}{- 63}$

$x = 5 - 5 z$

$y = - 2 + 4 z$

Simplify the left side.

Cancel the common factor of $- 63$ .

Cancel the common factor.

$\frac{- 63 z}{- 63} = \frac{- 63}{- 63}$

$x = 5 - 5 z$

$y = - 2 + 4 z$

Divide $z$ by $1$ .

$z = \frac{- 63}{- 63}$

$x = 5 - 5 z$

$y = - 2 + 4 z$

$z = \frac{- 63}{- 63}$

$x = 5 - 5 z$

$y = - 2 + 4 z$

$z = \frac{- 63}{- 63}$

$x = 5 - 5 z$

$y = - 2 + 4 z$

Simplify the right side.

Divide $- 63$ by $- 63$ .

$z = 1$

$x = 5 - 5 z$

$y = - 2 + 4 z$

$z = 1$

$x = 5 - 5 z$

$y = - 2 + 4 z$

$z = 1$

$x = 5 - 5 z$

$y = - 2 + 4 z$

$z = 1$

$x = 5 - 5 z$

$y = - 2 + 4 z$

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

Replace all occurrences of $z$ in $x = 5 - 5 z$ with $1$ .

$x = 5 - 5 \cdot 1$

$z = 1$

$y = - 2 + 4 z$

Simplify the right side.

Simplify $5 - 5 \cdot 1$ .

Multiply $- 5$ by $1$ .

$x = 5 - 5$

$z = 1$

$y = - 2 + 4 z$

Subtract $5$ from $5$ .

$x = 0$

$z = 1$

$y = - 2 + 4 z$

$x = 0$

$z = 1$

$y = - 2 + 4 z$

$x = 0$

$z = 1$

$y = - 2 + 4 z$

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

$y = - 2 + 4 (1)$

$x = 0$

$z = 1$

Simplify the right side.

Simplify $- 2 + 4 (1)$ .

Multiply $4$ by $1$ .

$y = - 2 + 4$

$x = 0$

$z = 1$

Add $- 2$ and $4$ .

$y = 2$

$x = 0$

$z = 1$

$y = 2$

$x = 0$

$z = 1$

$y = 2$

$x = 0$

$z = 1$

$y = 2$

$x = 0$

$z = 1$

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 2, 1)$

The result can be shown in multiple forms.

Point Form:

$(0, 2, 1)$

Equation Form:

$x = 0, y = 2, z = 1$

Solve the System of Equations x+y+z=3 13x+2z=2 -x-5z=-5

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