Solve the System of Equations x+9y+z=20 x+10y-2z=18 3x+27y+2z=58

$x + 9 y + z = 20$ $x + 10 y - 2 z = 18$ $3 x + 27 y + 2 z = 58$

Step 1

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

Subtract $9 y$ from both sides of the equation.

$x + z = 20 - 9 y$

$x + 10 y - 2 z = 18$

$3 x + 27 y + 2 z = 58$

Subtract $z$ from both sides of the equation.

$x = 20 - 9 y - z$

$x + 10 y - 2 z = 18$

$3 x + 27 y + 2 z = 58$

$x = 20 - 9 y - z$

$x + 10 y - 2 z = 18$

$3 x + 27 y + 2 z = 58$

Step 2

Replace all occurrences of $x$ with $20 - 9 y - z$ in each equation.

Replace all occurrences of $x$ in $x + 10 y - 2 z = 18$ with $20 - 9 y - z$ .

$(20 - 9 y - z) + 10 y - 2 z = 18$

$x = 20 - 9 y - z$

$3 x + 27 y + 2 z = 58$

Simplify the left side.

Simplify $(20 - 9 y - z) + 10 y - 2 z$ .

Add $- 9 y$ and $10 y$ .

$20 + y - z - 2 z = 18$

$x = 20 - 9 y - z$

$3 x + 27 y + 2 z = 58$

Subtract $2 z$ from $- z$ .

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$3 x + 27 y + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$3 x + 27 y + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$3 x + 27 y + 2 z = 58$

Replace all occurrences of $x$ in $3 x + 27 y + 2 z = 58$ with $20 - 9 y - z$ .

$3 (20 - 9 y - z) + 27 y + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Simplify the left side.

Simplify $3 (20 - 9 y - z) + 27 y + 2 z$ .

Simplify each term.

Apply the distributive property.

$3 \cdot 20 + 3 (- 9 y) + 3 (- z) + 27 y + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Simplify.

Multiply $3$ by $20$ .

$60 + 3 (- 9 y) + 3 (- z) + 27 y + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Multiply $- 9$ by $3$ .

$60 - 27 y + 3 (- z) + 27 y + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Multiply $- 1$ by $3$ .

$60 - 27 y - 3 z + 27 y + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$60 - 27 y - 3 z + 27 y + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$60 - 27 y - 3 z + 27 y + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Simplify by adding terms.

Combine the opposite terms in $60 - 27 y - 3 z + 27 y + 2 z$ .

Add $- 27 y$ and $27 y$ .

$60 - 3 z + 0 + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Add $60 - 3 z$ and $0$ .

$60 - 3 z + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$60 - 3 z + 2 z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Add $- 3 z$ and $2 z$ .

$60 - z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$60 - z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$60 - z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$60 - z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$60 - z = 58$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Step 3

Solve for $z$ in $60 - z = 58$ .

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

Subtract $60$ from both sides of the equation.

$- z = 58 - 60$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Subtract $60$ from $58$ .

$- z = - 2$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$- z = - 2$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Divide each term in $- z = - 2$ by $- 1$ and simplify.

Divide each term in $- z = - 2$ by $- 1$ .

$\frac{- z}{- 1} = \frac{- 2}{- 1}$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{z}{1} = \frac{- 2}{- 1}$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Divide $z$ by $1$ .

$z = \frac{- 2}{- 1}$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$z = \frac{- 2}{- 1}$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$z = 2$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$z = 2$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$z = 2$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

$z = 2$

$20 + y - 3 z = 18$

$x = 20 - 9 y - z$

Step 4

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

Replace all occurrences of $z$ in $20 + y - 3 z = 18$ with $2$ .

$20 + y - 3 \cdot 2 = 18$

$z = 2$

$x = 20 - 9 y - z$

Simplify the left side.

Simplify $20 + y - 3 \cdot 2$ .

Multiply $- 3$ by $2$ .

$20 + y - 6 = 18$

$z = 2$

$x = 20 - 9 y - z$

Subtract $6$ from $20$ .

$y + 14 = 18$

$z = 2$

$x = 20 - 9 y - z$

$y + 14 = 18$

$z = 2$

$x = 20 - 9 y - z$

$y + 14 = 18$

$z = 2$

$x = 20 - 9 y - z$

Replace all occurrences of $z$ in $x = 20 - 9 y - z$ with $2$ .

$x = 20 - 9 y - (2)$

$y + 14 = 18$

$z = 2$

Simplify the right side.

Simplify $20 - 9 y - (2)$ .

Multiply $- 1$ by $2$ .

$x = 20 - 9 y - 2$

$y + 14 = 18$

$z = 2$

Subtract $2$ from $20$ .

$x = - 9 y + 18$

$y + 14 = 18$

$z = 2$

$x = - 9 y + 18$

$y + 14 = 18$

$z = 2$

$x = - 9 y + 18$

$y + 14 = 18$

$z = 2$

$x = - 9 y + 18$

$y + 14 = 18$

$z = 2$

Step 5

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

Subtract $14$ from both sides of the equation.

$y = 18 - 14$

$x = - 9 y + 18$

$z = 2$

Subtract $14$ from $18$ .

$y = 4$

$x = - 9 y + 18$

$z = 2$

$y = 4$

$x = - 9 y + 18$

$z = 2$

Step 6

Replace all occurrences of $y$ with $4$ in each equation.

Replace all occurrences of $y$ in $x = - 9 y + 18$ with $4$ .

$x = - 9 \cdot 4 + 18$

$y = 4$

$z = 2$

Simplify the right side.

Simplify $- 9 \cdot 4 + 18$ .

Multiply $- 9$ by $4$ .

$x = - 36 + 18$

$y = 4$

$z = 2$

Add $- 36$ and $18$ .

$x = - 18$

$y = 4$

$z = 2$

$x = - 18$

$y = 4$

$z = 2$

$x = - 18$

$y = 4$

$z = 2$

$x = - 18$

$y = 4$

$z = 2$

Step 7

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

$(- 18, 4, 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 18, 4, 2)$

Equation Form:

$x = - 18, y = 4, z = 2$

Solve the System of Equations x+9y+z=20 x+10y-2z=18 3x+27y+2z=58

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