Solve the System of Equations 2x+3y=18 -9y+z=-23 5x-4z=50

$2 x + 3 y = 18$ $- 9 y + z = - 23$ $5 x - 4 z = 50$

Add $9 y$ to both sides of the equation.

$z = - 23 + 9 y$

$2 x + 3 y = 18$

$5 x - 4 z = 50$

Replace all occurrences of $z$ with $- 23 + 9 y$ in each equation.

Replace all occurrences of $z$ in $5 x - 4 z = 50$ with $- 23 + 9 y$ .

$5 x - 4 (- 23 + 9 y) = 50$

$z = - 23 + 9 y$

$2 x + 3 y = 18$

Simplify the left side.

Simplify each term.

Apply the distributive property.

$5 x - 4 \cdot - 23 - 4 (9 y) = 50$

$z = - 23 + 9 y$

$2 x + 3 y = 18$

Multiply $- 4$ by $- 23$ .

$5 x + 92 - 4 (9 y) = 50$

$z = - 23 + 9 y$

$2 x + 3 y = 18$

Multiply $9$ by $- 4$ .

$5 x + 92 - 36 y = 50$

$z = - 23 + 9 y$

$2 x + 3 y = 18$

$5 x + 92 - 36 y = 50$

$z = - 23 + 9 y$

$2 x + 3 y = 18$

$5 x + 92 - 36 y = 50$

$z = - 23 + 9 y$

$2 x + 3 y = 18$

$5 x + 92 - 36 y = 50$

$z = - 23 + 9 y$

$2 x + 3 y = 18$

Reorder $- 23$ and $9 y$ .

$z = 9 y - 23$

$5 x + 92 - 36 y = 50$

$2 x + 3 y = 18$

Solve for $x$ in $5 x + 92 - 36 y = 50$ .

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

Subtract $92$ from both sides of the equation.

$5 x - 36 y = 50 - 92$

$z = 9 y - 23$

$2 x + 3 y = 18$

Add $36 y$ to both sides of the equation.

$5 x = 50 - 92 + 36 y$

$z = 9 y - 23$

$2 x + 3 y = 18$

Subtract $92$ from $50$ .

$5 x = - 42 + 36 y$

$z = 9 y - 23$

$2 x + 3 y = 18$

$5 x = - 42 + 36 y$

$z = 9 y - 23$

$2 x + 3 y = 18$

Divide each term in $5 x = - 42 + 36 y$ by $5$ and simplify.

Divide each term in $5 x = - 42 + 36 y$ by $5$ .

$\frac{5 x}{5} = \frac{- 42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$2 x + 3 y = 18$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$2 x + 3 y = 18$

Divide $x$ by $1$ .

$x = \frac{- 42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$2 x + 3 y = 18$

$x = \frac{- 42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$2 x + 3 y = 18$

$x = \frac{- 42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$2 x + 3 y = 18$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$2 x + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$2 x + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$2 x + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$2 x + 3 y = 18$

Replace all occurrences of $x$ with $- \frac{42}{5} + \frac{36 y}{5}$ in each equation.

Replace all occurrences of $x$ in $2 x + 3 y = 18$ with $- \frac{42}{5} + \frac{36 y}{5}$ .

$2 (- \frac{42}{5} + \frac{36 y}{5}) + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Simplify the left side.

Simplify $2 (- \frac{42}{5} + \frac{36 y}{5}) + 3 y$ .

Simplify each term.

Apply the distributive property.

$2 (- \frac{42}{5}) + 2 (\frac{36 y}{5}) + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Multiply $2 (- \frac{42}{5})$ .

Multiply $- 1$ by $2$ .

$- 2 (\frac{42}{5}) + 2 (\frac{36 y}{5}) + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Combine $- 2$ and $\frac{42}{5}$ .

$\frac{- 2 \cdot 42}{5} + 2 (\frac{36 y}{5}) + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Multiply $- 2$ by $42$ .

$\frac{- 84}{5} + 2 (\frac{36 y}{5}) + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$\frac{- 84}{5} + 2 (\frac{36 y}{5}) + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Multiply $2 \frac{36 y}{5}$ .

Combine $2$ and $\frac{36 y}{5}$ .

$\frac{- 84}{5} + \frac{2 (36 y)}{5} + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Multiply $36$ by $2$ .

$\frac{- 84}{5} + \frac{72 y}{5} + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$\frac{- 84}{5} + \frac{72 y}{5} + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Move the negative in front of the fraction.

$- \frac{84}{5} + \frac{72 y}{5} + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$- \frac{84}{5} + \frac{72 y}{5} + 3 y = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

To write $3 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{84}{5} + \frac{72 y}{5} + 3 y \cdot \frac{5}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

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

$- \frac{84}{5} + \frac{72 y}{5} + \frac{3 y \cdot 5}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Combine the numerators over the common denominator.

$- \frac{84}{5} + \frac{72 y + 3 y \cdot 5}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Combine the numerators over the common denominator.

$\frac{- 84 + 72 y + 3 y \cdot 5}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Multiply $5$ by $3$ .

$\frac{- 84 + 72 y + 15 y}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Add $72 y$ and $15 y$ .

$\frac{- 84 + 87 y}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Factor $3$ out of $- 84 + 87 y$ .

Factor $3$ out of $- 84$ .

$\frac{3 (- 28) + 87 y}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Factor $3$ out of $87 y$ .

$\frac{3 (- 28) + 3 (29 y)}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Factor $3$ out of $3 (- 28) + 3 (29 y)$ .

$\frac{3 (- 28 + 29 y)}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$\frac{3 (- 28 + 29 y)}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Rewrite $- 28$ as $- 1 (28)$ .

$\frac{3 (- 1 \cdot 28 + 29 y)}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Factor $- 1$ out of $29 y$ .

$\frac{3 (- 1 \cdot 28 - (- 29 y))}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Factor $- 1$ out of $- 1 (28) - (- 29 y)$ .

$\frac{3 (- 1 (28 - 29 y))}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Move the negative in front of the fraction.

$- \frac{3 (28 - 29 y)}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$- \frac{3 (28 - 29 y)}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$- \frac{3 (28 - 29 y)}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$- \frac{3 (28 - 29 y)}{5} = 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Solve for $y$ in $- \frac{3 (28 - 29 y)}{5} = 18$ .

Multiply both sides of the equation by $- \frac{5}{3}$ .

$- \frac{5}{3} \cdot (- \frac{3 (28 - 29 y)}{5}) = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{5}{3} (- \frac{3 (28 - 29 y)}{5})$ .

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$\frac{- 5}{3} \cdot (- \frac{3 (28 - 29 y)}{5}) = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Move the leading negative in $- \frac{3 (28 - 29 y)}{5}$ into the numerator.

$\frac{- 5}{3} \cdot \frac{- 3 (28 - 29 y)}{5} = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{3} \cdot \frac{- 3 (28 - 29 y)}{5} = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Cancel the common factor.

$\frac{5 \cdot - 1}{3} \cdot \frac{- 3 (28 - 29 y)}{5} = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Rewrite the expression.

$\frac{- 1}{3} \cdot (- 3 (28 - 29 y)) = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$\frac{- 1}{3} \cdot (- 3 (28 - 29 y)) = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Cancel the common factor of $3$ .

Factor $3$ out of $- 3 (28 - 29 y)$ .

$\frac{- 1}{3} \cdot (3 (- (28 - 29 y))) = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Cancel the common factor.

$\frac{- 1}{3} \cdot (3 (- (28 - 29 y))) = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Rewrite the expression.

$28 - 29 y = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$28 - 29 y = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (28 - 29 y) = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Multiply $28 - 29 y$ by $1$ .

$28 - 29 y = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$28 - 29 y = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$28 - 29 y = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$28 - 29 y = - \frac{5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Simplify the right side.

Simplify $- \frac{5}{3} \cdot 18$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$28 - 29 y = \frac{- 5}{3} \cdot 18$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Factor $3$ out of $18$ .

$28 - 29 y = \frac{- 5}{3} \cdot (3 (6))$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Cancel the common factor.

$28 - 29 y = \frac{- 5}{3} \cdot (3 \cdot 6)$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Rewrite the expression.

$28 - 29 y = - 5 \cdot 6$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$28 - 29 y = - 5 \cdot 6$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Multiply $- 5$ by $6$ .

$28 - 29 y = - 30$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$28 - 29 y = - 30$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$28 - 29 y = - 30$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$28 - 29 y = - 30$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

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

Subtract $28$ from both sides of the equation.

$- 29 y = - 30 - 28$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Subtract $28$ from $- 30$ .

$- 29 y = - 58$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$- 29 y = - 58$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Divide each term in $- 29 y = - 58$ by $- 29$ and simplify.

Divide each term in $- 29 y = - 58$ by $- 29$ .

$\frac{- 29 y}{- 29} = \frac{- 58}{- 29}$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Simplify the left side.

Cancel the common factor of $- 29$ .

Cancel the common factor.

$\frac{- 29 y}{- 29} = \frac{- 58}{- 29}$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Divide $y$ by $1$ .

$y = \frac{- 58}{- 29}$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$y = \frac{- 58}{- 29}$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$y = \frac{- 58}{- 29}$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Simplify the right side.

Divide $- 58$ by $- 29$ .

$y = 2$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$y = 2$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$y = 2$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

$y = 2$

$x = - \frac{42}{5} + \frac{36 y}{5}$

$z = 9 y - 23$

Replace all occurrences of $y$ with $2$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{42}{5} + \frac{36 y}{5}$ with $2$ .

$x = - \frac{42}{5} + \frac{36 (2)}{5}$

$y = 2$

$z = 9 y - 23$

Simplify the right side.

Simplify $- \frac{42}{5} + \frac{36 (2)}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{- 42 + 36 (2)}{5}$

$y = 2$

$z = 9 y - 23$

Simplify the expression.

Multiply $36$ by $2$ .

$x = \frac{- 42 + 72}{5}$

$y = 2$

$z = 9 y - 23$

Add $- 42$ and $72$ .

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

$y = 2$

$z = 9 y - 23$

Divide $30$ by $5$ .

$x = 6$

$y = 2$

$z = 9 y - 23$

$x = 6$

$y = 2$

$z = 9 y - 23$

$x = 6$

$y = 2$

$z = 9 y - 23$

$x = 6$

$y = 2$

$z = 9 y - 23$

Replace all occurrences of $y$ in $z = 9 y - 23$ with $2$ .

$z = 9 (2) - 23$

$x = 6$

$y = 2$

Simplify the right side.

Simplify $9 (2) - 23$ .

Multiply $9$ by $2$ .

$z = 18 - 23$

$x = 6$

$y = 2$

Subtract $23$ from $18$ .

$z = - 5$

$x = 6$

$y = 2$

$z = - 5$

$x = 6$

$y = 2$

$z = - 5$

$x = 6$

$y = 2$

$z = - 5$

$x = 6$

$y = 2$

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

$(6, 2, - 5)$

The result can be shown in multiple forms.

Point Form:

$(6, 2, - 5)$

Equation Form:

$x = 6, y = 2, z = - 5$

Solve the System of Equations 2x+3y=18 -9y+z=-23 5x-4z=50

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