Finite Math Examples

$- 2 x + 2 y - 6 z = - 10$ , $x - y + 3 z = 5$

Step 1

Solve the equation for $y$ .

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

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

$2 y - 6 z = - 10 + 2 x$

$x - y + 3 z = 5$

Add $6 z$ to both sides of the equation.

$2 y = - 10 + 2 x + 6 z$

$x - y + 3 z = 5$

$2 y = - 10 + 2 x + 6 z$

$x - y + 3 z = 5$

Divide each term in $2 y = - 10 + 2 x + 6 z$ by $2$ and simplify.

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Divide each term in $2 y = - 10 + 2 x + 6 z$ by $2$ .

$\frac{2 y}{2} = \frac{- 10}{2} + \frac{2 x}{2} + \frac{6 z}{2}$

$x - y + 3 z = 5$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 y}{2} = \frac{- 10}{2} + \frac{2 x}{2} + \frac{6 z}{2}$

$x - y + 3 z = 5$

Divide $y$ by $1$ .

$y = \frac{- 10}{2} + \frac{2 x}{2} + \frac{6 z}{2}$

$x - y + 3 z = 5$

$y = \frac{- 10}{2} + \frac{2 x}{2} + \frac{6 z}{2}$

$x - y + 3 z = 5$

$y = \frac{- 10}{2} + \frac{2 x}{2} + \frac{6 z}{2}$

$x - y + 3 z = 5$

Simplify the right side.

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Simplify each term.

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Divide $- 10$ by $2$ .

$y = - 5 + \frac{2 x}{2} + \frac{6 z}{2}$

$x - y + 3 z = 5$

Cancel the common factor of $2$ .

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Cancel the common factor.

$y = - 5 + \frac{2 x}{2} + \frac{6 z}{2}$

$x - y + 3 z = 5$

Divide $x$ by $1$ .

$y = - 5 + x + \frac{6 z}{2}$

$x - y + 3 z = 5$

$y = - 5 + x + \frac{6 z}{2}$

$x - y + 3 z = 5$

Cancel the common factor of $6$ and $2$ .

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Factor $2$ out of $6 z$ .

$y = - 5 + x + \frac{2 (3 z)}{2}$

$x - y + 3 z = 5$

Cancel the common factors.

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Factor $2$ out of $2$ .

$y = - 5 + x + \frac{2 (3 z)}{2 (1)}$

$x - y + 3 z = 5$

Cancel the common factor.

$y = - 5 + x + \frac{2 (3 z)}{2 \cdot 1}$

$x - y + 3 z = 5$

Rewrite the expression.

$y = - 5 + x + \frac{3 z}{1}$

$x - y + 3 z = 5$

Divide $3 z$ by $1$ .

$y = - 5 + x + 3 z$

$x - y + 3 z = 5$

$y = - 5 + x + 3 z$

$x - y + 3 z = 5$

$y = - 5 + x + 3 z$

$x - y + 3 z = 5$

$y = - 5 + x + 3 z$

$x - y + 3 z = 5$

$y = - 5 + x + 3 z$

$x - y + 3 z = 5$

$y = - 5 + x + 3 z$

$x - y + 3 z = 5$

$y = - 5 + x + 3 z$

$x - y + 3 z = 5$

Step 2

Solve the equation for $z$ .

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Simplify $x - (- 5 + x + 3 z) + 3 z$ .

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Simplify each term.

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Apply the distributive property.

$x + 5 - x - (3 z) + 3 z = 5$

$y = - 5 + x + 3 z$

Simplify.

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Multiply $- 1$ by $- 5$ .

$x + 5 - x - (3 z) + 3 z = 5$

$y = - 5 + x + 3 z$

Multiply $3$ by $- 1$ .

$x + 5 - x - 3 z + 3 z = 5$

$y = - 5 + x + 3 z$

$x + 5 - x - 3 z + 3 z = 5$

$y = - 5 + x + 3 z$

$x + 5 - x - 3 z + 3 z = 5$

$y = - 5 + x + 3 z$

Combine the opposite terms in $x + 5 - x - 3 z + 3 z$ .

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Subtract $x$ from $x$ .

$0 + 5 - 3 z + 3 z = 5$

$y = - 5 + x + 3 z$

Add $0$ and $5$ .

$5 - 3 z + 3 z = 5$

$y = - 5 + x + 3 z$

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

$5 + 0 = 5$

$y = - 5 + x + 3 z$

Add $5$ and $0$ .

$5 = 5$

$y = - 5 + x + 3 z$

$5 = 5$

$y = - 5 + x + 3 z$

$5 = 5$

$y = - 5 + x + 3 z$

Since $5 = 5$ , the equation will always be true.

Always true

$y = - 5 + x + 3 z$

Always true

$y = - 5 + x + 3 z$

Step 3

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

$y = - 5 + x + 3 z$

Always true

Step 4

Simplify the right side.

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Move $- 5$ .

$y = x + 3 z - 5$

Always true

$y = x + 3 z - 5$

Always true