Precalculus Examples

$2 x + 3 y + z = 9$ , $x + 2 y + 3 z = 6$ , $3 x + y + 2 z = 8$

Step 1

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

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Step 1.1

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

$3 y + z = 9 - 2 x$

$x + 2 y + 3 z = 6$

$3 x + y + 2 z = 8$

Step 1.2

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

$z = 9 - 2 x - 3 y$

$x + 2 y + 3 z = 6$

$3 x + y + 2 z = 8$

$z = 9 - 2 x - 3 y$

$x + 2 y + 3 z = 6$

$3 x + y + 2 z = 8$

Step 2

Replace all occurrences of $z$ with $9 - 2 x - 3 y$ in each equation.

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Step 2.1

Replace all occurrences of $z$ in $x + 2 y + 3 z = 6$ with $9 - 2 x - 3 y$ .

$x + 2 y + 3 (9 - 2 x - 3 y) = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x + 2 y + 3 (9 - 2 x - 3 y)$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$x + 2 y + 3 \cdot 9 + 3 (- 2 x) + 3 (- 3 y) = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $9$ .

$x + 2 y + 27 + 3 (- 2 x) + 3 (- 3 y) = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

Step 2.2.1.1.2.2

Multiply $- 2$ by $3$ .

$x + 2 y + 27 - 6 x + 3 (- 3 y) = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

Step 2.2.1.1.2.3

Multiply $- 3$ by $3$ .

$x + 2 y + 27 - 6 x - 9 y = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

$x + 2 y + 27 - 6 x - 9 y = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

$x + 2 y + 27 - 6 x - 9 y = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $6 x$ from $x$ .

$- 5 x + 2 y + 27 - 9 y = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

Step 2.2.1.2.2

Subtract $9 y$ from $2 y$ .

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 2 z = 8$

Step 2.3

Replace all occurrences of $z$ in $3 x + y + 2 z = 8$ with $9 - 2 x - 3 y$ .

$3 x + y + 2 (9 - 2 x - 3 y) = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $3 x + y + 2 (9 - 2 x - 3 y)$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$3 x + y + 2 \cdot 9 + 2 (- 2 x) + 2 (- 3 y) = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $9$ .

$3 x + y + 18 + 2 (- 2 x) + 2 (- 3 y) = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 2.4.1.1.2.2

Multiply $- 2$ by $2$ .

$3 x + y + 18 - 4 x + 2 (- 3 y) = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 2.4.1.1.2.3

Multiply $- 3$ by $2$ .

$3 x + y + 18 - 4 x - 6 y = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 18 - 4 x - 6 y = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$3 x + y + 18 - 4 x - 6 y = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $4 x$ from $3 x$ .

$- x + y + 18 - 6 y = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 2.4.1.2.2

Subtract $6 y$ from $y$ .

$- x - 5 y + 18 = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$- x - 5 y + 18 = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$- x - 5 y + 18 = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$- x - 5 y + 18 = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$- x - 5 y + 18 = 8$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3

Solve for $x$ in $- x - 5 y + 18 = 8$ .

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Step 3.1

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

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Step 3.1.1

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

$- x + 18 = 8 + 5 y$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3.1.2

Subtract $18$ from both sides of the equation.

$- x = 8 + 5 y - 18$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3.1.3

Subtract $18$ from $8$ .

$- x = 5 y - 10$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$- x = 5 y - 10$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3.2

Divide each term in $- x = 5 y - 10$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x = 5 y - 10$ by $- 1$ .

$\frac{- x}{- 1} = \frac{5 y}{- 1} + \frac{- 10}{- 1}$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{5 y}{- 1} + \frac{- 10}{- 1}$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3.2.2.2

Divide $x$ by $1$ .

$x = \frac{5 y}{- 1} + \frac{- 10}{- 1}$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$x = \frac{5 y}{- 1} + \frac{- 10}{- 1}$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Move the negative one from the denominator of $\frac{5 y}{- 1}$ .

$x = - 1 \cdot (5 y) + \frac{- 10}{- 1}$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3.2.3.1.2

Rewrite $- 1 \cdot (5 y)$ as $- (5 y)$ .

$x = - (5 y) + \frac{- 10}{- 1}$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3.2.3.1.3

Multiply $5$ by $- 1$ .

$x = - 5 y + \frac{- 10}{- 1}$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 3.2.3.1.4

Divide $- 10$ by $- 1$ .

$x = - 5 y + 10$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$x = - 5 y + 10$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$x = - 5 y + 10$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$x = - 5 y + 10$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

$x = - 5 y + 10$

$- 5 x - 7 y + 27 = 6$

$z = 9 - 2 x - 3 y$

Step 4

Replace all occurrences of $x$ with $- 5 y + 10$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $- 5 x - 7 y + 27 = 6$ with $- 5 y + 10$ .

$- 5 (- 5 y + 10) - 7 y + 27 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 5 (- 5 y + 10) - 7 y + 27$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 5 (- 5 y) - 5 \cdot 10 - 7 y + 27 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

Step 4.2.1.1.2

Multiply $- 5$ by $- 5$ .

$25 y - 5 \cdot 10 - 7 y + 27 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

Step 4.2.1.1.3

Multiply $- 5$ by $10$ .

$25 y - 50 - 7 y + 27 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

$25 y - 50 - 7 y + 27 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $7 y$ from $25 y$ .

$18 y - 50 + 27 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

Step 4.2.1.2.2

Add $- 50$ and $27$ .

$18 y - 23 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

$18 y - 23 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

$18 y - 23 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

$18 y - 23 = 6$

$x = - 5 y + 10$

$z = 9 - 2 x - 3 y$

Step 4.3

Replace all occurrences of $x$ in $z = 9 - 2 x - 3 y$ with $- 5 y + 10$ .

$z = 9 - 2 (- 5 y + 10) - 3 y$

$18 y - 23 = 6$

$x = - 5 y + 10$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $9 - 2 (- 5 y + 10) - 3 y$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$z = 9 - 2 (- 5 y) - 2 \cdot 10 - 3 y$

$18 y - 23 = 6$

$x = - 5 y + 10$

Step 4.4.1.1.2

Multiply $- 5$ by $- 2$ .

$z = 9 + 10 y - 2 \cdot 10 - 3 y$

$18 y - 23 = 6$

$x = - 5 y + 10$

Step 4.4.1.1.3

Multiply $- 2$ by $10$ .

$z = 9 + 10 y - 20 - 3 y$

$18 y - 23 = 6$

$x = - 5 y + 10$

$z = 9 + 10 y - 20 - 3 y$

$18 y - 23 = 6$

$x = - 5 y + 10$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $20$ from $9$ .

$z = 10 y - 11 - 3 y$

$18 y - 23 = 6$

$x = - 5 y + 10$

Step 4.4.1.2.2

Subtract $3 y$ from $10 y$ .

$z = 7 y - 11$

$18 y - 23 = 6$

$x = - 5 y + 10$

$z = 7 y - 11$

$18 y - 23 = 6$

$x = - 5 y + 10$

$z = 7 y - 11$

$18 y - 23 = 6$

$x = - 5 y + 10$

$z = 7 y - 11$

$18 y - 23 = 6$

$x = - 5 y + 10$

$z = 7 y - 11$

$18 y - 23 = 6$

$x = - 5 y + 10$

Step 5

Solve for $y$ in $18 y - 23 = 6$ .

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Step 5.1

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

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Step 5.1.1

Add $23$ to both sides of the equation.

$18 y = 6 + 23$

$z = 7 y - 11$

$x = - 5 y + 10$

Step 5.1.2

Add $6$ and $23$ .

$18 y = 29$

$z = 7 y - 11$

$x = - 5 y + 10$

$18 y = 29$

$z = 7 y - 11$

$x = - 5 y + 10$

Step 5.2

Divide each term in $18 y = 29$ by $18$ and simplify.

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Step 5.2.1

Divide each term in $18 y = 29$ by $18$ .

$\frac{18 y}{18} = \frac{29}{18}$

$z = 7 y - 11$

$x = - 5 y + 10$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $18$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{18 y}{18} = \frac{29}{18}$

$z = 7 y - 11$

$x = - 5 y + 10$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{29}{18}$

$z = 7 y - 11$

$x = - 5 y + 10$

$y = \frac{29}{18}$

$z = 7 y - 11$

$x = - 5 y + 10$

$y = \frac{29}{18}$

$z = 7 y - 11$

$x = - 5 y + 10$

$y = \frac{29}{18}$

$z = 7 y - 11$

$x = - 5 y + 10$

$y = \frac{29}{18}$

$z = 7 y - 11$

$x = - 5 y + 10$

Step 6

Replace all occurrences of $y$ with $\frac{29}{18}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $z = 7 y - 11$ with $\frac{29}{18}$ .

$z = 7 (\frac{29}{18}) - 11$

$y = \frac{29}{18}$

$x = - 5 y + 10$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $7 (\frac{29}{18}) - 11$ .

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Step 6.2.1.1

Multiply $7 (\frac{29}{18})$ .

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Step 6.2.1.1.1

Combine $7$ and $\frac{29}{18}$ .

$z = \frac{7 \cdot 29}{18} - 11$

$y = \frac{29}{18}$

$x = - 5 y + 10$

Step 6.2.1.1.2

Multiply $7$ by $29$ .

$z = \frac{203}{18} - 11$

$y = \frac{29}{18}$

$x = - 5 y + 10$

$z = \frac{203}{18} - 11$

$y = \frac{29}{18}$

$x = - 5 y + 10$

Step 6.2.1.2

To write $- 11$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$z = \frac{203}{18} - 11 \cdot \frac{18}{18}$

$y = \frac{29}{18}$

$x = - 5 y + 10$

Step 6.2.1.3

Combine $- 11$ and $\frac{18}{18}$ .

$z = \frac{203}{18} + \frac{- 11 \cdot 18}{18}$

$y = \frac{29}{18}$

$x = - 5 y + 10$

Step 6.2.1.4

Combine the numerators over the common denominator.

$z = \frac{203 - 11 \cdot 18}{18}$

$y = \frac{29}{18}$

$x = - 5 y + 10$

Step 6.2.1.5

Simplify the numerator.

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Step 6.2.1.5.1

Multiply $- 11$ by $18$ .

$z = \frac{203 - 198}{18}$

$y = \frac{29}{18}$

$x = - 5 y + 10$

Step 6.2.1.5.2

Subtract $198$ from $203$ .

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = - 5 y + 10$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = - 5 y + 10$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = - 5 y + 10$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = - 5 y + 10$

Step 6.3

Replace all occurrences of $y$ in $x = - 5 y + 10$ with $\frac{29}{18}$ .

$x = - 5 (\frac{29}{18}) + 10$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 5 (\frac{29}{18}) + 10$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Multiply $- 5 (\frac{29}{18})$ .

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Step 6.4.1.1.1.1

Combine $- 5$ and $\frac{29}{18}$ .

$x = \frac{- 5 \cdot 29}{18} + 10$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

Step 6.4.1.1.1.2

Multiply $- 5$ by $29$ .

$x = \frac{- 145}{18} + 10$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = \frac{- 145}{18} + 10$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

Step 6.4.1.1.2

Move the negative in front of the fraction.

$x = - \frac{145}{18} + 10$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = - \frac{145}{18} + 10$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

Step 6.4.1.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$x = - \frac{145}{18} + 10 \cdot \frac{18}{18}$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

Step 6.4.1.3

Combine $10$ and $\frac{18}{18}$ .

$x = - \frac{145}{18} + \frac{10 \cdot 18}{18}$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

Step 6.4.1.4

Combine the numerators over the common denominator.

$x = \frac{- 145 + 10 \cdot 18}{18}$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

Step 6.4.1.5

Simplify the numerator.

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Step 6.4.1.5.1

Multiply $10$ by $18$ .

$x = \frac{- 145 + 180}{18}$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

Step 6.4.1.5.2

Add $- 145$ and $180$ .

$x = \frac{35}{18}$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = \frac{35}{18}$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = \frac{35}{18}$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = \frac{35}{18}$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

$x = \frac{35}{18}$

$z = \frac{5}{18}$

$y = \frac{29}{18}$

Step 7

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

$(\frac{35}{18}, \frac{29}{18}, \frac{5}{18})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{35}{18}, \frac{29}{18}, \frac{5}{18})$

Equation Form:

$x = \frac{35}{18}, y = \frac{29}{18}, z = \frac{5}{18}$