Solve the System of Equations 3x+2y=13 3x+2y+z=13 2x+y+3z=9

$3 x + 2 y = 13$ $3 x + 2 y + z = 13$ $2 x + y + 3 z = 9$

Step 1

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

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

$2 y + z = 13 - 3 x$

$3 x + 2 y = 13$

$2 x + y + 3 z = 9$

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

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$2 x + y + 3 z = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$2 x + y + 3 z = 9$

Step 2

Replace all occurrences of $z$ with $13 - 3 x - 2 y$ in each equation.

Replace all occurrences of $z$ in $2 x + y + 3 z = 9$ with $13 - 3 x - 2 y$ .

$2 x + y + 3 (13 - 3 x - 2 y) = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$2 x + y + 3 \cdot 13 + 3 (- 3 x) + 3 (- 2 y) = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Simplify.

Multiply $3$ by $13$ .

$2 x + y + 39 + 3 (- 3 x) + 3 (- 2 y) = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Multiply $- 3$ by $3$ .

$2 x + y + 39 - 9 x + 3 (- 2 y) = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Multiply $- 2$ by $3$ .

$2 x + y + 39 - 9 x - 6 y = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$2 x + y + 39 - 9 x - 6 y = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$2 x + y + 39 - 9 x - 6 y = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Simplify by adding terms.

Subtract $9 x$ from $2 x$ .

$- 7 x + y + 39 - 6 y = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Subtract $6 y$ from $y$ .

$- 7 x - 5 y + 39 = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$- 7 x - 5 y + 39 = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$- 7 x - 5 y + 39 = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$- 7 x - 5 y + 39 = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$- 7 x - 5 y + 39 = 9$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Step 3

Solve for $x$ in $- 7 x - 5 y + 39 = 9$ .

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

Add $5 y$ to both sides of the equation.

$- 7 x + 39 = 9 + 5 y$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Subtract $39$ from both sides of the equation.

$- 7 x = 9 + 5 y - 39$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Subtract $39$ from $9$ .

$- 7 x = 5 y - 30$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$- 7 x = 5 y - 30$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Divide each term in $- 7 x = 5 y - 30$ by $- 7$ and simplify.

Divide each term in $- 7 x = 5 y - 30$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{5 y}{- 7} + \frac{- 30}{- 7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{5 y}{- 7} + \frac{- 30}{- 7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Divide $x$ by $1$ .

$x = \frac{5 y}{- 7} + \frac{- 30}{- 7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$x = \frac{5 y}{- 7} + \frac{- 30}{- 7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$x = \frac{5 y}{- 7} + \frac{- 30}{- 7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{5 y}{7} + \frac{- 30}{- 7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Dividing two negative values results in a positive value.

$x = - \frac{5 y}{7} + \frac{30}{7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$z = 13 - 3 x - 2 y$

$3 x + 2 y = 13$

Step 4

Replace all occurrences of $x$ with $- \frac{5 y}{7} + \frac{30}{7}$ in each equation.

Replace all occurrences of $x$ in $z = 13 - 3 x - 2 y$ with $- \frac{5 y}{7} + \frac{30}{7}$ .

$z = 13 - 3 (- \frac{5 y}{7} + \frac{30}{7}) - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Simplify the right side.

Simplify $13 - 3 (- \frac{5 y}{7} + \frac{30}{7}) - 2 y$ .

Simplify each term.

Apply the distributive property.

$z = 13 - 3 (- \frac{5 y}{7}) - 3 (\frac{30}{7}) - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Multiply $- 3 (- \frac{5 y}{7})$ .

Multiply $- 1$ by $- 3$ .

$z = 13 + 3 (\frac{5 y}{7}) - 3 (\frac{30}{7}) - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Combine $3$ and $\frac{5 y}{7}$ .

$z = 13 + \frac{3 (5 y)}{7} - 3 (\frac{30}{7}) - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Multiply $5$ by $3$ .

$z = 13 + \frac{15 y}{7} - 3 (\frac{30}{7}) - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

$z = 13 + \frac{15 y}{7} - 3 (\frac{30}{7}) - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Multiply $- 3 (\frac{30}{7})$ .

Combine $- 3$ and $\frac{30}{7}$ .

$z = 13 + \frac{15 y}{7} + \frac{- 3 \cdot 30}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Multiply $- 3$ by $30$ .

$z = 13 + \frac{15 y}{7} + \frac{- 90}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

$z = 13 + \frac{15 y}{7} + \frac{- 90}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Move the negative in front of the fraction.

$z = 13 + \frac{15 y}{7} - \frac{90}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

$z = 13 + \frac{15 y}{7} - \frac{90}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

To write $13$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$z = \frac{15 y}{7} + 13 \cdot \frac{7}{7} - \frac{90}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Combine $13$ and $\frac{7}{7}$ .

$z = \frac{15 y}{7} + \frac{13 \cdot 7}{7} - \frac{90}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Combine the numerators over the common denominator.

$z = \frac{15 y}{7} + \frac{13 \cdot 7 - 90}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Simplify the numerator.

Multiply $13$ by $7$ .

$z = \frac{15 y}{7} + \frac{91 - 90}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Subtract $90$ from $91$ .

$z = \frac{15 y}{7} + \frac{1}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

$z = \frac{15 y}{7} + \frac{1}{7} - 2 y$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$z = \frac{15 y}{7} - 2 y \cdot \frac{7}{7} + \frac{1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Combine $- 2 y$ and $\frac{7}{7}$ .

$z = \frac{15 y}{7} + \frac{- 2 y \cdot 7}{7} + \frac{1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Combine the numerators over the common denominator.

$z = \frac{15 y - 2 y \cdot 7}{7} + \frac{1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Combine the numerators over the common denominator.

$z = \frac{15 y - 2 y \cdot 7 + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Multiply $7$ by $- 2$ .

$z = \frac{15 y - 14 y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Subtract $14 y$ from $15 y$ .

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$3 x + 2 y = 13$

Replace all occurrences of $x$ in $3 x + 2 y = 13$ with $- \frac{5 y}{7} + \frac{30}{7}$ .

$3 (- \frac{5 y}{7} + \frac{30}{7}) + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Simplify the left side.

Simplify $3 (- \frac{5 y}{7} + \frac{30}{7}) + 2 y$ .

Simplify each term.

Apply the distributive property.

$3 (- \frac{5 y}{7}) + 3 (\frac{30}{7}) + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Multiply $3 (- \frac{5 y}{7})$ .

Multiply $- 1$ by $3$ .

$- 3 \frac{5 y}{7} + 3 (\frac{30}{7}) + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Combine $- 3$ and $\frac{5 y}{7}$ .

$\frac{- 3 (5 y)}{7} + 3 (\frac{30}{7}) + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Multiply $5$ by $- 3$ .

$\frac{- 15 y}{7} + 3 (\frac{30}{7}) + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$\frac{- 15 y}{7} + 3 (\frac{30}{7}) + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Multiply $3 (\frac{30}{7})$ .

Combine $3$ and $\frac{30}{7}$ .

$\frac{- 15 y}{7} + \frac{3 \cdot 30}{7} + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Multiply $3$ by $30$ .

$\frac{- 15 y}{7} + \frac{90}{7} + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$\frac{- 15 y}{7} + \frac{90}{7} + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Move the negative in front of the fraction.

$- \frac{15 y}{7} + \frac{90}{7} + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$- \frac{15 y}{7} + \frac{90}{7} + 2 y = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{15 y}{7} + 2 y \cdot \frac{7}{7} + \frac{90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Combine $2 y$ and $\frac{7}{7}$ .

$- \frac{15 y}{7} + \frac{2 y \cdot 7}{7} + \frac{90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Combine the numerators over the common denominator.

$\frac{- 15 y + 2 y \cdot 7}{7} + \frac{90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Combine the numerators over the common denominator.

$\frac{- 15 y + 2 y \cdot 7 + 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Multiply $7$ by $2$ .

$\frac{- 15 y + 14 y + 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Add $- 15 y$ and $14 y$ .

$\frac{- y + 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Factor $- 1$ out of $- y$ .

$\frac{- (y) + 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Rewrite $90$ as $- 1 (- 90)$ .

$\frac{- (y) - 1 \cdot - 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Factor $- 1$ out of $- (y) - 1 (- 90)$ .

$\frac{- (y - 90)}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Simplify the expression.

Rewrite $- (y - 90)$ as $- 1 (y - 90)$ .

$\frac{- 1 (y - 90)}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Move the negative in front of the fraction.

$- \frac{y - 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$- \frac{y - 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$- \frac{y - 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$- \frac{y - 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$- \frac{y - 90}{7} = 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Step 5

Solve for $y$ in $- \frac{y - 90}{7} = 13$ .

Multiply both sides of the equation by $- 7$ .

$- 7 (- \frac{y - 90}{7}) = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 7 (- \frac{y - 90}{7})$ .

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{y - 90}{7}$ into the numerator.

$- 7 \frac{- (y - 90)}{7} = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Factor $7$ out of $- 7$ .

$7 (- 1) (\frac{- (y - 90)}{7}) = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Cancel the common factor.

$7 \cdot (- 1 \frac{- (y - 90)}{7}) = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Rewrite the expression.

$y - 90 = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$y - 90 = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (y - 90) = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Multiply $y - 90$ by $1$ .

$y - 90 = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$y - 90 = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$y - 90 = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$y - 90 = - 7 \cdot 13$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Simplify the right side.

Multiply $- 7$ by $13$ .

$y - 90 = - 91$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$y - 90 = - 91$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$y - 90 = - 91$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

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

Add $90$ to both sides of the equation.

$y = - 91 + 90$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Add $- 91$ and $90$ .

$y = - 1$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$y = - 1$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$y = - 1$

$z = \frac{y + 1}{7}$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Step 6

Replace all occurrences of $y$ with $- 1$ in each equation.

Replace all occurrences of $y$ in $z = \frac{y + 1}{7}$ with $- 1$ .

$z = \frac{(- 1) + 1}{7}$

$y = - 1$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Simplify the right side.

Simplify $\frac{(- 1) + 1}{7}$ .

Add $- 1$ and $1$ .

$z = \frac{0}{7}$

$y = - 1$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Divide $0$ by $7$ .

$z = 0$

$y = - 1$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$z = 0$

$y = - 1$

$x = - \frac{5 y}{7} + \frac{30}{7}$

$z = 0$

$y = - 1$

$x = - \frac{5 y}{7} + \frac{30}{7}$

Replace all occurrences of $y$ in $x = - \frac{5 y}{7} + \frac{30}{7}$ with $- 1$ .

$x = - \frac{5 (- 1)}{7} + \frac{30}{7}$

$z = 0$

$y = - 1$

Simplify the right side.

Simplify $- \frac{5 (- 1)}{7} + \frac{30}{7}$ .

Combine the numerators over the common denominator.

$x = \frac{- 5 \cdot - 1 + 30}{7}$

$z = 0$

$y = - 1$

Simplify the expression.

Multiply $- 5$ by $- 1$ .

$x = \frac{5 + 30}{7}$

$z = 0$

$y = - 1$

Add $5$ and $30$ .

$x = \frac{35}{7}$

$z = 0$

$y = - 1$

Divide $35$ by $7$ .

$x = 5$

$z = 0$

$y = - 1$

$x = 5$

$z = 0$

$y = - 1$

$x = 5$

$z = 0$

$y = - 1$

$x = 5$

$z = 0$

$y = - 1$

$x = 5$

$z = 0$

$y = - 1$

Step 7

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

$(5, - 1, 0)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(5, - 1, 0)$

Equation Form:

$x = 5, y = - 1, z = 0$

Solve the System of Equations 3x+2y=13 3x+2y+z=13 2x+y+3z=9

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