Finite Math Examples

$x + 3 y + 3 z = 3$ , $x + 2 y + 3 z = 5$ , $x + 3 y + 2 z = 5$

Step 1

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

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Step 1.1

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

$x + 3 z = 3 - 3 y$

$x + 2 y + 3 z = 5$

$x + 3 y + 2 z = 5$

Step 1.2

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

$x = 3 - 3 y - 3 z$

$x + 2 y + 3 z = 5$

$x + 3 y + 2 z = 5$

$x = 3 - 3 y - 3 z$

$x + 2 y + 3 z = 5$

$x + 3 y + 2 z = 5$

Step 2

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

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Step 2.1

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

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

$x = 3 - 3 y - 3 z$

$x + 3 y + 2 z = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Combine the opposite terms in $(3 - 3 y - 3 z) + 2 y + 3 z$ .

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Step 2.2.1.1.1

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

$(3 - 3 y) + 2 y + 0 = 5$

$x = 3 - 3 y - 3 z$

$x + 3 y + 2 z = 5$

Step 2.2.1.1.2

Add $(3 - 3 y) + 2 y$ and $0$ .

$(3 - 3 y) + 2 y = 5$

$x = 3 - 3 y - 3 z$

$x + 3 y + 2 z = 5$

$(3 - 3 y) + 2 y = 5$

$x = 3 - 3 y - 3 z$

$x + 3 y + 2 z = 5$

Step 2.2.1.2

Add $- 3 y$ and $2 y$ .

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$x + 3 y + 2 z = 5$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$x + 3 y + 2 z = 5$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$x + 3 y + 2 z = 5$

Step 2.3

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

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

$3 - y = 5$

$x = 3 - 3 y - 3 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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Step 2.4.1.1

Combine the opposite terms in $(3 - 3 y - 3 z) + 3 y + 2 z$ .

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Step 2.4.1.1.1

Add $- 3 y$ and $3 y$ .

$3 - 3 z + 0 + 2 z = 5$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

Step 2.4.1.1.2

Add $3 - 3 z$ and $0$ .

$3 - 3 z + 2 z = 5$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$3 - 3 z + 2 z = 5$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

Step 2.4.1.2

Add $- 3 z$ and $2 z$ .

$3 - z = 5$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$3 - z = 5$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$3 - z = 5$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$3 - z = 5$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

Step 3

Solve for $z$ in $3 - z = 5$ .

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Step 3.1

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

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Step 3.1.1

Subtract $3$ from both sides of the equation.

$- z = 5 - 3$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

Step 3.1.2

Subtract $3$ from $5$ .

$- z = 2$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$- z = 2$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

Step 3.2

Divide each term in $- z = 2$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- z = 2$ by $- 1$ .

$\frac{- z}{- 1} = \frac{2}{- 1}$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

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.

$\frac{z}{1} = \frac{2}{- 1}$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

Step 3.2.2.2

Divide $z$ by $1$ .

$z = \frac{2}{- 1}$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$z = \frac{2}{- 1}$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $2$ by $- 1$ .

$z = - 2$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$z = - 2$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$z = - 2$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

$z = - 2$

$3 - y = 5$

$x = 3 - 3 y - 3 z$

Step 4

Replace all occurrences of $z$ with $- 2$ in each equation.

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Step 4.1

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

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Step 4.1.1

Subtract $3$ from both sides of the equation.

$- y = 5 - 3$

$z = - 2$

$x = 3 - 3 y - 3 z$

Step 4.1.2

Subtract $3$ from $5$ .

$- y = 2$

$z = - 2$

$x = 3 - 3 y - 3 z$

$- y = 2$

$z = - 2$

$x = 3 - 3 y - 3 z$

Step 4.2

Divide each term in $- y = 2$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- y = 2$ by $- 1$ .

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

$z = - 2$

$x = 3 - 3 y - 3 z$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

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

$z = - 2$

$x = 3 - 3 y - 3 z$

Step 4.2.2.2

Divide $y$ by $1$ .

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

$z = - 2$

$x = 3 - 3 y - 3 z$

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

$z = - 2$

$x = 3 - 3 y - 3 z$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $2$ by $- 1$ .

$y = - 2$

$z = - 2$

$x = 3 - 3 y - 3 z$

$y = - 2$

$z = - 2$

$x = 3 - 3 y - 3 z$

$y = - 2$

$z = - 2$

$x = 3 - 3 y - 3 z$

Step 4.3

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

$x = 3 - 3 y - 3 \cdot - 2$

$y = - 2$

$z = - 2$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $3 - 3 y - 3 \cdot - 2$ .

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Step 4.4.1.1

Multiply $- 3$ by $- 2$ .

$x = 3 - 3 y + 6$

$y = - 2$

$z = - 2$

Step 4.4.1.2

Add $3$ and $6$ .

$x = - 3 y + 9$

$y = - 2$

$z = - 2$

$x = - 3 y + 9$

$y = - 2$

$z = - 2$

$x = - 3 y + 9$

$y = - 2$

$z = - 2$

$x = - 3 y + 9$

$y = - 2$

$z = - 2$

Step 5

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 5.1

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

$x = - 3 \cdot - 2 + 9$

$y = - 2$

$z = - 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 3 \cdot - 2 + 9$ .

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Step 5.2.1.1

Multiply $- 3$ by $- 2$ .

$x = 6 + 9$

$y = - 2$

$z = - 2$

Step 5.2.1.2

Add $6$ and $9$ .

$x = 15$

$y = - 2$

$z = - 2$

$x = 15$

$y = - 2$

$z = - 2$

$x = 15$

$y = - 2$

$z = - 2$

$x = 15$

$y = - 2$

$z = - 2$

Step 6

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

$(15, - 2, - 2)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(15, - 2, - 2)$

Equation Form:

$x = 15, y = - 2, z = - 2$