Finite Math Examples

$2 x + y + 6 z = 3$ , $x - y + 4 z = 1$ , $3 x + 2 y - 2 z = 2$

Step 1

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

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Step 1.1

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

$y + 6 z = 3 - 2 x$

$x - y + 4 z = 1$

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

Step 1.2

Subtract $6 z$ from both sides of the equation.

$y = 3 - 2 x - 6 z$

$x - y + 4 z = 1$

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

$y = 3 - 2 x - 6 z$

$x - y + 4 z = 1$

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

Step 2

Replace all occurrences of $y$ with $3 - 2 x - 6 z$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x - y + 4 z = 1$ with $3 - 2 x - 6 z$ .

$x - (3 - 2 x - 6 z) + 4 z = 1$

$y = 3 - 2 x - 6 z$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x - (3 - 2 x - 6 z) + 4 z$ .

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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.

$x - 1 \cdot 3 - (- 2 x) - (- 6 z) + 4 z = 1$

$y = 3 - 2 x - 6 z$

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

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $3$ .

$x - 3 - (- 2 x) - (- 6 z) + 4 z = 1$

$y = 3 - 2 x - 6 z$

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

Step 2.2.1.1.2.2

Multiply $- 2$ by $- 1$ .

$x - 3 + 2 x - (- 6 z) + 4 z = 1$

$y = 3 - 2 x - 6 z$

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

Step 2.2.1.1.2.3

Multiply $- 6$ by $- 1$ .

$x - 3 + 2 x + 6 z + 4 z = 1$

$y = 3 - 2 x - 6 z$

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

$x - 3 + 2 x + 6 z + 4 z = 1$

$y = 3 - 2 x - 6 z$

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

$x - 3 + 2 x + 6 z + 4 z = 1$

$y = 3 - 2 x - 6 z$

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

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $x$ and $2 x$ .

$3 x - 3 + 6 z + 4 z = 1$

$y = 3 - 2 x - 6 z$

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

Step 2.2.1.2.2

Add $6 z$ and $4 z$ .

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

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

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

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

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

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

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

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

Step 2.3

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

$3 x + 2 (3 - 2 x - 6 z) - 2 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $3 x + 2 (3 - 2 x - 6 z) - 2 z$ .

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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.

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

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $3$ .

$3 x + 6 + 2 (- 2 x) + 2 (- 6 z) - 2 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 2.4.1.1.2.2

Multiply $- 2$ by $2$ .

$3 x + 6 - 4 x + 2 (- 6 z) - 2 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 2.4.1.1.2.3

Multiply $- 6$ by $2$ .

$3 x + 6 - 4 x - 12 z - 2 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$3 x + 6 - 4 x - 12 z - 2 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$3 x + 6 - 4 x - 12 z - 2 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $4 x$ from $3 x$ .

$- x + 6 - 12 z - 2 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 2.4.1.2.2

Subtract $2 z$ from $- 12 z$ .

$- x + 6 - 14 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$- x + 6 - 14 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$- x + 6 - 14 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$- x + 6 - 14 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$- x + 6 - 14 z = 2$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 3

Solve for $x$ in $- x + 6 - 14 z = 2$ .

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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

Subtract $6$ from both sides of the equation.

$- x - 14 z = 2 - 6$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 3.1.2

Add $14 z$ to both sides of the equation.

$- x = 2 - 6 + 14 z$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 3.1.3

Subtract $6$ from $2$ .

$- x = - 4 + 14 z$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$- x = - 4 + 14 z$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 3.2

Divide each term in $- x = - 4 + 14 z$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x = - 4 + 14 z$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1} + \frac{14 z}{- 1}$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

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.

$\frac{x}{1} = \frac{- 4}{- 1} + \frac{14 z}{- 1}$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1} + \frac{14 z}{- 1}$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$x = \frac{- 4}{- 1} + \frac{14 z}{- 1}$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

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

Divide $- 4$ by $- 1$ .

$x = 4 + \frac{14 z}{- 1}$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 3.2.3.1.2

Move the negative one from the denominator of $\frac{14 z}{- 1}$ .

$x = 4 - 1 \cdot (14 z)$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 3.2.3.1.3

Rewrite $- 1 \cdot (14 z)$ as $- (14 z)$ .

$x = 4 - (14 z)$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 3.2.3.1.4

Multiply $14$ by $- 1$ .

$x = 4 - 14 z$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$x = 4 - 14 z$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$x = 4 - 14 z$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$x = 4 - 14 z$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

$x = 4 - 14 z$

$3 x - 3 + 10 z = 1$

$y = 3 - 2 x - 6 z$

Step 4

Replace all occurrences of $x$ with $4 - 14 z$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $3 x - 3 + 10 z = 1$ with $4 - 14 z$ .

$3 (4 - 14 z) - 3 + 10 z = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $3 (4 - 14 z) - 3 + 10 z$ .

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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.

$3 \cdot 4 + 3 (- 14 z) - 3 + 10 z = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

Step 4.2.1.1.2

Multiply $3$ by $4$ .

$12 + 3 (- 14 z) - 3 + 10 z = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

Step 4.2.1.1.3

Multiply $- 14$ by $3$ .

$12 - 42 z - 3 + 10 z = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

$12 - 42 z - 3 + 10 z = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $3$ from $12$ .

$- 42 z + 9 + 10 z = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

Step 4.2.1.2.2

Add $- 42 z$ and $10 z$ .

$- 32 z + 9 = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

$y = 3 - 2 x - 6 z$

Step 4.3

Replace all occurrences of $x$ in $y = 3 - 2 x - 6 z$ with $4 - 14 z$ .

$y = 3 - 2 (4 - 14 z) - 6 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $3 - 2 (4 - 14 z) - 6 z$ .

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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.

$y = 3 - 2 \cdot 4 - 2 (- 14 z) - 6 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

Step 4.4.1.1.2

Multiply $- 2$ by $4$ .

$y = 3 - 8 - 2 (- 14 z) - 6 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

Step 4.4.1.1.3

Multiply $- 14$ by $- 2$ .

$y = 3 - 8 + 28 z - 6 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

$y = 3 - 8 + 28 z - 6 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $8$ from $3$ .

$y = - 5 + 28 z - 6 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

Step 4.4.1.2.2

Subtract $6 z$ from $28 z$ .

$y = - 5 + 22 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

$y = - 5 + 22 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

$y = - 5 + 22 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

$y = - 5 + 22 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

$y = - 5 + 22 z$

$- 32 z + 9 = 1$

$x = 4 - 14 z$

Step 5

Solve for $z$ in $- 32 z + 9 = 1$ .

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Step 5.1

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

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Step 5.1.1

Subtract $9$ from both sides of the equation.

$- 32 z = 1 - 9$

$y = - 5 + 22 z$

$x = 4 - 14 z$

Step 5.1.2

Subtract $9$ from $1$ .

$- 32 z = - 8$

$y = - 5 + 22 z$

$x = 4 - 14 z$

$- 32 z = - 8$

$y = - 5 + 22 z$

$x = 4 - 14 z$

Step 5.2

Divide each term in $- 32 z = - 8$ by $- 32$ and simplify.

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Step 5.2.1

Divide each term in $- 32 z = - 8$ by $- 32$ .

$\frac{- 32 z}{- 32} = \frac{- 8}{- 32}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 32$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 32 z}{- 32} = \frac{- 8}{- 32}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 8}{- 32}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

$z = \frac{- 8}{- 32}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

$z = \frac{- 8}{- 32}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $- 8$ and $- 32$ .

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Step 5.2.3.1.1

Factor $- 8$ out of $- 8$ .

$z = \frac{- 8 \cdot 1}{- 32}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $- 8$ out of $- 32$ .

$z = \frac{- 8 \cdot 1}{- 8 \cdot 4}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

Step 5.2.3.1.2.2

Cancel the common factor.

$z = \frac{- 8 \cdot 1}{- 8 \cdot 4}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

Step 5.2.3.1.2.3

Rewrite the expression.

$z = \frac{1}{4}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

$z = \frac{1}{4}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

$z = \frac{1}{4}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

$z = \frac{1}{4}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

$z = \frac{1}{4}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

$z = \frac{1}{4}$

$y = - 5 + 22 z$

$x = 4 - 14 z$

Step 6

Replace all occurrences of $z$ with $\frac{1}{4}$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $y = - 5 + 22 z$ with $\frac{1}{4}$ .

$y = - 5 + 22 (\frac{1}{4})$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 5 + 22 (\frac{1}{4})$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1.1

Factor $2$ out of $22$ .

$y = - 5 + 2 (11) (\frac{1}{4})$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2.1.1.1.2

Factor $2$ out of $4$ .

$y = - 5 + 2 \cdot (11 (\frac{1}{2 \cdot 2}))$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2.1.1.1.3

Cancel the common factor.

$y = - 5 + 2 \cdot (11 (\frac{1}{2 \cdot 2}))$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2.1.1.1.4

Rewrite the expression.

$y = - 5 + 11 (\frac{1}{2})$

$z = \frac{1}{4}$

$x = 4 - 14 z$

$y = - 5 + 11 (\frac{1}{2})$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2.1.1.2

Combine $11$ and $\frac{1}{2}$ .

$y = - 5 + \frac{11}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

$y = - 5 + \frac{11}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2.1.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - 5 \cdot \frac{2}{2} + \frac{11}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2.1.3

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

$y = \frac{- 5 \cdot 2}{2} + \frac{11}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2.1.4

Combine the numerators over the common denominator.

$y = \frac{- 5 \cdot 2 + 11}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2.1.5

Simplify the numerator.

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Step 6.2.1.5.1

Multiply $- 5$ by $2$ .

$y = \frac{- 10 + 11}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.2.1.5.2

Add $- 10$ and $11$ .

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = 4 - 14 z$

Step 6.3

Replace all occurrences of $z$ in $x = 4 - 14 z$ with $\frac{1}{4}$ .

$x = 4 - 14 (\frac{1}{4})$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $4 - 14 (\frac{1}{4})$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Cancel the common factor of $2$ .

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Step 6.4.1.1.1.1

Factor $2$ out of $- 14$ .

$x = 4 + 2 (- 7) (\frac{1}{4})$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.1.1.2

Factor $2$ out of $4$ .

$x = 4 + 2 \cdot (- 7 \frac{1}{2 \cdot 2})$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.1.1.3

Cancel the common factor.

$x = 4 + 2 \cdot (- 7 \frac{1}{2 \cdot 2})$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.1.1.4

Rewrite the expression.

$x = 4 - 7 (\frac{1}{2})$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = 4 - 7 (\frac{1}{2})$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.1.2

Combine $- 7$ and $\frac{1}{2}$ .

$x = 4 + \frac{- 7}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.1.3

Move the negative in front of the fraction.

$x = 4 - \frac{7}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = 4 - \frac{7}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 4 \cdot \frac{2}{2} - \frac{7}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.3

Combine $4$ and $\frac{2}{2}$ .

$x = \frac{4 \cdot 2}{2} - \frac{7}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.4

Combine the numerators over the common denominator.

$x = \frac{4 \cdot 2 - 7}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.5

Simplify the numerator.

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Step 6.4.1.5.1

Multiply $4$ by $2$ .

$x = \frac{8 - 7}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 6.4.1.5.2

Subtract $7$ from $8$ .

$x = \frac{1}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = \frac{1}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = \frac{1}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = \frac{1}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

$x = \frac{1}{2}$

$y = \frac{1}{2}$

$z = \frac{1}{4}$

Step 7

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

$(\frac{1}{2}, \frac{1}{2}, \frac{1}{4})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{2}, \frac{1}{2}, \frac{1}{4})$

Equation Form:

$x = \frac{1}{2}, y = \frac{1}{2}, z = \frac{1}{4}$