Finite Math Examples

$3 x + 3 y + z = 3$ , $x - 3 y + 2 z = - 16$ , $8 x - 2 y + 3 z = - 23$

Step 1

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

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Step 1.1

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

$3 y + z = 3 - 3 x$

$x - 3 y + 2 z = - 16$

$8 x - 2 y + 3 z = - 23$

Step 1.2

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

$z = 3 - 3 x - 3 y$

$x - 3 y + 2 z = - 16$

$8 x - 2 y + 3 z = - 23$

$z = 3 - 3 x - 3 y$

$x - 3 y + 2 z = - 16$

$8 x - 2 y + 3 z = - 23$

Step 2

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

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Step 2.1

Replace all occurrences of $z$ in $x - 3 y + 2 z = - 16$ with $3 - 3 x - 3 y$ .

$x - 3 y + 2 (3 - 3 x - 3 y) = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$x - 3 y + 2 \cdot 3 + 2 (- 3 x) + 2 (- 3 y) = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $3$ .

$x - 3 y + 6 + 2 (- 3 x) + 2 (- 3 y) = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

Step 2.2.1.1.2.2

Multiply $- 3$ by $2$ .

$x - 3 y + 6 - 6 x + 2 (- 3 y) = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

Step 2.2.1.1.2.3

Multiply $- 3$ by $2$ .

$x - 3 y + 6 - 6 x - 6 y = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

$x - 3 y + 6 - 6 x - 6 y = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

$x - 3 y + 6 - 6 x - 6 y = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $6 x$ from $x$ .

$- 5 x - 3 y + 6 - 6 y = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

Step 2.2.1.2.2

Subtract $6 y$ from $- 3 y$ .

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 3 z = - 23$

Step 2.3

Replace all occurrences of $z$ in $8 x - 2 y + 3 z = - 23$ with $3 - 3 x - 3 y$ .

$8 x - 2 y + 3 (3 - 3 x - 3 y) = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $8 x - 2 y + 3 (3 - 3 x - 3 y)$ .

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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.

$8 x - 2 y + 3 \cdot 3 + 3 (- 3 x) + 3 (- 3 y) = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $3$ by $3$ .

$8 x - 2 y + 9 + 3 (- 3 x) + 3 (- 3 y) = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 2.4.1.1.2.2

Multiply $- 3$ by $3$ .

$8 x - 2 y + 9 - 9 x + 3 (- 3 y) = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 2.4.1.1.2.3

Multiply $- 3$ by $3$ .

$8 x - 2 y + 9 - 9 x - 9 y = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 9 - 9 x - 9 y = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$8 x - 2 y + 9 - 9 x - 9 y = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $9 x$ from $8 x$ .

$- x - 2 y + 9 - 9 y = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 2.4.1.2.2

Subtract $9 y$ from $- 2 y$ .

$- x - 11 y + 9 = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$- x - 11 y + 9 = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$- x - 11 y + 9 = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$- x - 11 y + 9 = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$- x - 11 y + 9 = - 23$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 3

Solve for $x$ in $- x - 11 y + 9 = - 23$ .

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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 $11 y$ to both sides of the equation.

$- x + 9 = - 23 + 11 y$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 3.1.2

Subtract $9$ from both sides of the equation.

$- x = - 23 + 11 y - 9$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 3.1.3

Subtract $9$ from $- 23$ .

$- x = 11 y - 32$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$- x = 11 y - 32$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 3.2

Divide each term in $- x = 11 y - 32$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x = 11 y - 32$ by $- 1$ .

$\frac{- x}{- 1} = \frac{11 y}{- 1} + \frac{- 32}{- 1}$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

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.

$\frac{x}{1} = \frac{11 y}{- 1} + \frac{- 32}{- 1}$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 3.2.2.2

Divide $x$ by $1$ .

$x = \frac{11 y}{- 1} + \frac{- 32}{- 1}$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$x = \frac{11 y}{- 1} + \frac{- 32}{- 1}$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

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 $\frac{11 y}{- 1}$ .

$x = - 1 \cdot (11 y) + \frac{- 32}{- 1}$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 3.2.3.1.2

Rewrite $- 1 \cdot (11 y)$ as $- (11 y)$ .

$x = - (11 y) + \frac{- 32}{- 1}$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 3.2.3.1.3

Multiply $11$ by $- 1$ .

$x = - 11 y + \frac{- 32}{- 1}$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 3.2.3.1.4

Divide $- 32$ by $- 1$ .

$x = - 11 y + 32$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$x = - 11 y + 32$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$x = - 11 y + 32$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$x = - 11 y + 32$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

$x = - 11 y + 32$

$- 5 x - 9 y + 6 = - 16$

$z = 3 - 3 x - 3 y$

Step 4

Replace all occurrences of $x$ with $- 11 y + 32$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $- 5 x - 9 y + 6 = - 16$ with $- 11 y + 32$ .

$- 5 (- 11 y + 32) - 9 y + 6 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 5 (- 11 y + 32) - 9 y + 6$ .

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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.

$- 5 (- 11 y) - 5 \cdot 32 - 9 y + 6 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

Step 4.2.1.1.2

Multiply $- 11$ by $- 5$ .

$55 y - 5 \cdot 32 - 9 y + 6 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

Step 4.2.1.1.3

Multiply $- 5$ by $32$ .

$55 y - 160 - 9 y + 6 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

$55 y - 160 - 9 y + 6 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $9 y$ from $55 y$ .

$46 y - 160 + 6 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

Step 4.2.1.2.2

Add $- 160$ and $6$ .

$46 y - 154 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

$46 y - 154 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

$46 y - 154 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

$46 y - 154 = - 16$

$x = - 11 y + 32$

$z = 3 - 3 x - 3 y$

Step 4.3

Replace all occurrences of $x$ in $z = 3 - 3 x - 3 y$ with $- 11 y + 32$ .

$z = 3 - 3 (- 11 y + 32) - 3 y$

$46 y - 154 = - 16$

$x = - 11 y + 32$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $3 - 3 (- 11 y + 32) - 3 y$ .

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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.

$z = 3 - 3 (- 11 y) - 3 \cdot 32 - 3 y$

$46 y - 154 = - 16$

$x = - 11 y + 32$

Step 4.4.1.1.2

Multiply $- 11$ by $- 3$ .

$z = 3 + 33 y - 3 \cdot 32 - 3 y$

$46 y - 154 = - 16$

$x = - 11 y + 32$

Step 4.4.1.1.3

Multiply $- 3$ by $32$ .

$z = 3 + 33 y - 96 - 3 y$

$46 y - 154 = - 16$

$x = - 11 y + 32$

$z = 3 + 33 y - 96 - 3 y$

$46 y - 154 = - 16$

$x = - 11 y + 32$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $96$ from $3$ .

$z = 33 y - 93 - 3 y$

$46 y - 154 = - 16$

$x = - 11 y + 32$

Step 4.4.1.2.2

Subtract $3 y$ from $33 y$ .

$z = 30 y - 93$

$46 y - 154 = - 16$

$x = - 11 y + 32$

$z = 30 y - 93$

$46 y - 154 = - 16$

$x = - 11 y + 32$

$z = 30 y - 93$

$46 y - 154 = - 16$

$x = - 11 y + 32$

$z = 30 y - 93$

$46 y - 154 = - 16$

$x = - 11 y + 32$

$z = 30 y - 93$

$46 y - 154 = - 16$

$x = - 11 y + 32$

Step 5

Solve for $y$ in $46 y - 154 = - 16$ .

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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 $154$ to both sides of the equation.

$46 y = - 16 + 154$

$z = 30 y - 93$

$x = - 11 y + 32$

Step 5.1.2

Add $- 16$ and $154$ .

$46 y = 138$

$z = 30 y - 93$

$x = - 11 y + 32$

$46 y = 138$

$z = 30 y - 93$

$x = - 11 y + 32$

Step 5.2

Divide each term in $46 y = 138$ by $46$ and simplify.

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Step 5.2.1

Divide each term in $46 y = 138$ by $46$ .

$\frac{46 y}{46} = \frac{138}{46}$

$z = 30 y - 93$

$x = - 11 y + 32$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $46$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{46 y}{46} = \frac{138}{46}$

$z = 30 y - 93$

$x = - 11 y + 32$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{138}{46}$

$z = 30 y - 93$

$x = - 11 y + 32$

$y = \frac{138}{46}$

$z = 30 y - 93$

$x = - 11 y + 32$

$y = \frac{138}{46}$

$z = 30 y - 93$

$x = - 11 y + 32$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $138$ by $46$ .

$y = 3$

$z = 30 y - 93$

$x = - 11 y + 32$

$y = 3$

$z = 30 y - 93$

$x = - 11 y + 32$

$y = 3$

$z = 30 y - 93$

$x = - 11 y + 32$

$y = 3$

$z = 30 y - 93$

$x = - 11 y + 32$

Step 6

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

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Step 6.1

Replace all occurrences of $y$ in $z = 30 y - 93$ with $3$ .

$z = 30 (3) - 93$

$y = 3$

$x = - 11 y + 32$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $30 (3) - 93$ .

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Step 6.2.1.1

Multiply $30$ by $3$ .

$z = 90 - 93$

$y = 3$

$x = - 11 y + 32$

Step 6.2.1.2

Subtract $93$ from $90$ .

$z = - 3$

$y = 3$

$x = - 11 y + 32$

$z = - 3$

$y = 3$

$x = - 11 y + 32$

$z = - 3$

$y = 3$

$x = - 11 y + 32$

Step 6.3

Replace all occurrences of $y$ in $x = - 11 y + 32$ with $3$ .

$x = - 11 \cdot 3 + 32$

$z = - 3$

$y = 3$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 11 \cdot 3 + 32$ .

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Step 6.4.1.1

Multiply $- 11$ by $3$ .

$x = - 33 + 32$

$z = - 3$

$y = 3$

Step 6.4.1.2

Add $- 33$ and $32$ .

$x = - 1$

$z = - 3$

$y = 3$

$x = - 1$

$z = - 3$

$y = 3$

$x = - 1$

$z = - 3$

$y = 3$

$x = - 1$

$z = - 3$

$y = 3$

Step 7

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

$(- 1, 3, - 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 1, 3, - 3)$

Equation Form:

$x = - 1, y = 3, z = - 3$