Finite Math Examples

$- 4 x - 16 y - 2 z = 18$ , $- 2 x - 8 y - z = 9$

Step 1

Solve the equation for $x$ .

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

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Add $16 y$ to both sides of the equation.

$- 4 x - 2 z = 18 + 16 y$

$- 2 x - 8 y - z = 9$

Add $2 z$ to both sides of the equation.

$- 4 x = 18 + 16 y + 2 z$

$- 2 x - 8 y - z = 9$

$- 4 x = 18 + 16 y + 2 z$

$- 2 x - 8 y - z = 9$

Divide each term in $- 4 x = 18 + 16 y + 2 z$ by $- 4$ and simplify.

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Divide each term in $- 4 x = 18 + 16 y + 2 z$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{18}{- 4} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Simplify the left side.

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Cancel the common factor of $- 4$ .

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Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{18}{- 4} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Divide $x$ by $1$ .

$x = \frac{18}{- 4} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

$x = \frac{18}{- 4} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

$x = \frac{18}{- 4} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Simplify the right side.

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Simplify each term.

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Cancel the common factor of $18$ and $- 4$ .

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Factor $2$ out of $18$ .

$x = \frac{2 (9)}{- 4} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Cancel the common factors.

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Factor $2$ out of $- 4$ .

$x = \frac{2 \cdot 9}{2 \cdot - 2} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Cancel the common factor.

$x = \frac{2 \cdot 9}{2 \cdot - 2} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Rewrite the expression.

$x = \frac{9}{- 2} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

$x = \frac{9}{- 2} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

$x = \frac{9}{- 2} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Move the negative in front of the fraction.

$x = - \frac{9}{2} + \frac{16 y}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Cancel the common factor of $16$ and $- 4$ .

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Factor $4$ out of $16 y$ .

$x = - \frac{9}{2} + \frac{4 (4 y)}{- 4} + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Move the negative one from the denominator of $\frac{4 y}{- 1}$ .

$x = - \frac{9}{2} - 1 \cdot (4 y) + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

$x = - \frac{9}{2} - 1 \cdot (4 y) + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Rewrite $- 1 \cdot (4 y)$ as $- (4 y)$ .

$x = - \frac{9}{2} - (4 y) + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Multiply $4$ by $- 1$ .

$x = - \frac{9}{2} - 4 y + \frac{2 z}{- 4}$

$- 2 x - 8 y - z = 9$

Cancel the common factor of $2$ and $- 4$ .

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Factor $2$ out of $2 z$ .

$x = - \frac{9}{2} - 4 y + \frac{2 (z)}{- 4}$

$- 2 x - 8 y - z = 9$

Cancel the common factors.

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Factor $2$ out of $- 4$ .

$x = - \frac{9}{2} - 4 y + \frac{2 z}{2 \cdot - 2}$

$- 2 x - 8 y - z = 9$

Cancel the common factor.

$x = - \frac{9}{2} - 4 y + \frac{2 z}{2 \cdot - 2}$

$- 2 x - 8 y - z = 9$

Rewrite the expression.

$x = - \frac{9}{2} - 4 y + \frac{z}{- 2}$

$- 2 x - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y + \frac{z}{- 2}$

$- 2 x - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y + \frac{z}{- 2}$

$- 2 x - 8 y - z = 9$

Move the negative in front of the fraction.

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$- 2 x - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$- 2 x - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$- 2 x - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$- 2 x - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$- 2 x - 8 y - z = 9$

Step 2

Solve the equation for $z$ .

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Simplify $- 2 (- \frac{9}{2} - 4 y - \frac{z}{2}) - 8 y - z$ .

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Simplify each term.

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Apply the distributive property.

$- 2 (- \frac{9}{2}) - 2 (- 4 y) - 2 (- \frac{z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Simplify.

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Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{9}{2}$ into the numerator.

$- 2 (\frac{- 9}{2}) - 2 (- 4 y) - 2 (- \frac{z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- 9}{2}) - 2 (- 4 y) - 2 (- \frac{z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Cancel the common factor.

$2 \cdot (- 1 (\frac{- 9}{2})) - 2 (- 4 y) - 2 (- \frac{z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Rewrite the expression.

$- 1 \cdot - 9 - 2 (- 4 y) - 2 (- \frac{z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$- 1 \cdot - 9 - 2 (- 4 y) - 2 (- \frac{z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Multiply $- 1$ by $- 9$ .

$9 - 2 (- 4 y) - 2 (- \frac{z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Multiply $- 4$ by $- 2$ .

$9 + 8 y - 2 (- \frac{z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{z}{2}$ into the numerator.

$9 + 8 y - 2 \frac{- z}{2} - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Factor $2$ out of $- 2$ .

$9 + 8 y + 2 (- 1) (\frac{- z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Cancel the common factor.

$9 + 8 y + 2 \cdot (- 1 \frac{- z}{2}) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Rewrite the expression.

$9 + 8 y - 1 (- z) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$9 + 8 y - 1 (- z) - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Multiply $- 1$ by $- 1$ .

$9 + 8 y + 1 z - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Multiply $z$ by $1$ .

$9 + 8 y + z - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$9 + 8 y + z - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$9 + 8 y + z - 8 y - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Combine the opposite terms in $9 + 8 y + z - 8 y - z$ .

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Subtract $8 y$ from $8 y$ .

$9 + z + 0 - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Add $9 + z$ and $0$ .

$9 + z - z = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Subtract $z$ from $z$ .

$9 + 0 = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Add $9$ and $0$ .

$9 = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$9 = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

$9 = 9$

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Since $9 = 9$ , the equation will always be true.

Always true

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Always true

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Step 3

The simplified system is the arbitrary solution of the original system of equations.

$x = - \frac{9}{2} - 4 y - \frac{z}{2}$

Always true

Step 4

Simplify the right side.

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Move $- \frac{9}{2}$ .

$x = - 4 y - \frac{z}{2} - \frac{9}{2}$

Always true

$x = - 4 y - \frac{z}{2} - \frac{9}{2}$

Always true