Finite Math Examples

$3 x + 2 y + 6 z = 2$ , $x + y + 2 z = 1$ , $2 x + 2 y + 5 z = 3$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x + 2 z = 1 - y$

$3 x + 2 y + 6 z = 2$

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

Step 1.2

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

$x = 1 - y - 2 z$

$3 x + 2 y + 6 z = 2$

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

$x = 1 - y - 2 z$

$3 x + 2 y + 6 z = 2$

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

Step 2

Replace all occurrences of $x$ with $1 - y - 2 z$ in each equation.

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Step 2.1

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

$3 (1 - y - 2 z) + 2 y + 6 z = 2$

$x = 1 - y - 2 z$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (1 - y - 2 z) + 2 y + 6 z$ .

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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.

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

$x = 1 - y - 2 z$

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

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $1$ .

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

$x = 1 - y - 2 z$

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

Step 2.2.1.1.2.2

Multiply $- 1$ by $3$ .

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

$x = 1 - y - 2 z$

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

Step 2.2.1.1.2.3

Multiply $- 2$ by $3$ .

$3 - 3 y - 6 z + 2 y + 6 z = 2$

$x = 1 - y - 2 z$

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

$3 - 3 y - 6 z + 2 y + 6 z = 2$

$x = 1 - y - 2 z$

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

$3 - 3 y - 6 z + 2 y + 6 z = 2$

$x = 1 - y - 2 z$

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

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Combine the opposite terms in $3 - 3 y - 6 z + 2 y + 6 z$ .

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Step 2.2.1.2.1.1

Add $- 6 z$ and $6 z$ .

$3 - 3 y + 2 y + 0 = 2$

$x = 1 - y - 2 z$

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

Step 2.2.1.2.1.2

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

$3 - 3 y + 2 y = 2$

$x = 1 - y - 2 z$

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

$3 - 3 y + 2 y = 2$

$x = 1 - y - 2 z$

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

Step 2.2.1.2.2

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

$3 - y = 2$

$x = 1 - y - 2 z$

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

$3 - y = 2$

$x = 1 - y - 2 z$

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

$3 - y = 2$

$x = 1 - y - 2 z$

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

$3 - y = 2$

$x = 1 - y - 2 z$

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

Step 2.3

Replace all occurrences of $x$ in $2 x + 2 y + 5 z = 3$ with $1 - y - 2 z$ .

$2 (1 - y - 2 z) + 2 y + 5 z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (1 - y - 2 z) + 2 y + 5 z$ .

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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.

$2 \cdot 1 + 2 (- y) + 2 (- 2 z) + 2 y + 5 z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $1$ .

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

$3 - y = 2$

$x = 1 - y - 2 z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $2$ .

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

$3 - y = 2$

$x = 1 - y - 2 z$

Step 2.4.1.1.2.3

Multiply $- 2$ by $2$ .

$2 - 2 y - 4 z + 2 y + 5 z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

$2 - 2 y - 4 z + 2 y + 5 z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

$2 - 2 y - 4 z + 2 y + 5 z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Combine the opposite terms in $2 - 2 y - 4 z + 2 y + 5 z$ .

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Step 2.4.1.2.1.1

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

$2 - 4 z + 0 + 5 z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

Step 2.4.1.2.1.2

Add $2 - 4 z$ and $0$ .

$2 - 4 z + 5 z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

$2 - 4 z + 5 z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

Step 2.4.1.2.2

Add $- 4 z$ and $5 z$ .

$2 + z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

$2 + z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

$2 + z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

$2 + z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

$2 + z = 3$

$3 - y = 2$

$x = 1 - y - 2 z$

Step 3

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

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Step 3.1

Subtract $2$ from both sides of the equation.

$z = 3 - 2$

$3 - y = 2$

$x = 1 - y - 2 z$

Step 3.2

Subtract $2$ from $3$ .

$z = 1$

$3 - y = 2$

$x = 1 - y - 2 z$

$z = 1$

$3 - y = 2$

$x = 1 - y - 2 z$

Step 4

Replace all occurrences of $z$ with $1$ 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.

$- y = 2 - 3$

$z = 1$

$x = 1 - y - 2 z$

Step 4.1.2

Subtract $3$ from $2$ .

$- y = - 1$

$z = 1$

$x = 1 - y - 2 z$

$- y = - 1$

$z = 1$

$x = 1 - y - 2 z$

Step 4.2

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

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Step 4.2.1

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

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

$z = 1$

$x = 1 - y - 2 z$

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.

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

$z = 1$

$x = 1 - y - 2 z$

Step 4.2.2.2

Divide $y$ by $1$ .

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

$z = 1$

$x = 1 - y - 2 z$

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

$z = 1$

$x = 1 - y - 2 z$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 1$ by $- 1$ .

$y = 1$

$z = 1$

$x = 1 - y - 2 z$

$y = 1$

$z = 1$

$x = 1 - y - 2 z$

$y = 1$

$z = 1$

$x = 1 - y - 2 z$

Step 4.3

Replace all occurrences of $z$ in $x = 1 - y - 2 z$ with $1$ .

$x = 1 - y - 2 \cdot 1$

$y = 1$

$z = 1$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $1 - y - 2 \cdot 1$ .

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Step 4.4.1.1

Multiply $- 2$ by $1$ .

$x = 1 - y - 2$

$y = 1$

$z = 1$

Step 4.4.1.2

Subtract $2$ from $1$ .

$x = - y - 1$

$y = 1$

$z = 1$

$x = - y - 1$

$y = 1$

$z = 1$

$x = - y - 1$

$y = 1$

$z = 1$

$x = - y - 1$

$y = 1$

$z = 1$

Step 5

Replace all occurrences of $y$ with $1$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = - y - 1$ with $1$ .

$x = - (1) - 1$

$y = 1$

$z = 1$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- (1) - 1$ .

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Step 5.2.1.1

Multiply $- 1$ by $1$ .

$x = - 1 - 1$

$y = 1$

$z = 1$

Step 5.2.1.2

Subtract $1$ from $- 1$ .

$x = - 2$

$y = 1$

$z = 1$

$x = - 2$

$y = 1$

$z = 1$

$x = - 2$

$y = 1$

$z = 1$

$x = - 2$

$y = 1$

$z = 1$

Step 6

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

$(- 2, 1, 1)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 2, 1, 1)$

Equation Form:

$x = - 2, y = 1, z = 1$