Finite Math Examples

$x + y + z = 4$ , $x - y + 5 z = - 14$ , $5 x + y + z = 20$

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.

$x + z = 4 - y$

$x - y + 5 z = - 14$

$5 x + y + z = 20$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 4 - y - z$

$x - y + 5 z = - 14$

$5 x + y + z = 20$

$x = 4 - y - z$

$x - y + 5 z = - 14$

$5 x + y + z = 20$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x - y + 5 z = - 14$ with $4 - y - z$ .

$(4 - y - z) - y + 5 z = - 14$

$x = 4 - y - z$

$5 x + y + z = 20$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $(4 - y - z) - y + 5 z$ .

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Step 2.2.1.1

Subtract $y$ from $- y$ .

$4 - 2 y - z + 5 z = - 14$

$x = 4 - y - z$

$5 x + y + z = 20$

Step 2.2.1.2

Add $- z$ and $5 z$ .

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$5 x + y + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$5 x + y + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$5 x + y + z = 20$

Step 2.3

Replace all occurrences of $x$ in $5 x + y + z = 20$ with $4 - y - z$ .

$5 (4 - y - z) + y + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $5 (4 - y - z) + y + z$ .

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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.

$5 \cdot 4 + 5 (- y) + 5 (- z) + y + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $5$ by $4$ .

$20 + 5 (- y) + 5 (- z) + y + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $5$ .

$20 - 5 y + 5 (- z) + y + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $5$ .

$20 - 5 y - 5 z + y + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$20 - 5 y - 5 z + y + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$20 - 5 y - 5 z + y + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 5 y$ and $y$ .

$20 - 4 y - 5 z + z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 2.4.1.2.2

Add $- 5 z$ and $z$ .

$20 - 4 y - 4 z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$20 - 4 y - 4 z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$20 - 4 y - 4 z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$20 - 4 y - 4 z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$20 - 4 y - 4 z = 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3

Solve for $y$ in $20 - 4 y - 4 z = 20$ .

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Step 3.1

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

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Step 3.1.1

Subtract $20$ from both sides of the equation.

$- 4 y - 4 z = 20 - 20$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3.1.2

Add $4 z$ to both sides of the equation.

$- 4 y = 20 - 20 + 4 z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3.1.3

Subtract $20$ from $20$ .

$- 4 y = 0 + 4 z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3.1.4

Add $0$ and $4 z$ .

$- 4 y = 4 z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$- 4 y = 4 z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3.2

Divide each term in $- 4 y = 4 z$ by $- 4$ and simplify.

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Step 3.2.1

Divide each term in $- 4 y = 4 z$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{4 z}{- 4}$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{4 z}{- 4}$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4 z}{- 4}$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$y = \frac{4 z}{- 4}$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$y = \frac{4 z}{- 4}$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

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

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Step 3.2.3.1.1

Factor $4$ out of $4 z$ .

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

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3.2.3.1.2

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

$y = - 1 \cdot z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$y = - 1 \cdot z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 3.2.3.2

Rewrite $- 1 \cdot z$ as $- z$ .

$y = - z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$y = - z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$y = - z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

$y = - z$

$4 - 2 y + 4 z = - 14$

$x = 4 - y - z$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $4 - 2 y + 4 z = - 14$ with $- z$ .

$4 - 2 (- z) + 4 z = - 14$

$y = - z$

$x = 4 - y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 - 2 (- z) + 4 z$ .

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Step 4.2.1.1

Multiply $- 1$ by $- 2$ .

$4 + 2 z + 4 z = - 14$

$y = - z$

$x = 4 - y - z$

Step 4.2.1.2

Add $2 z$ and $4 z$ .

$4 + 6 z = - 14$

$y = - z$

$x = 4 - y - z$

$4 + 6 z = - 14$

$y = - z$

$x = 4 - y - z$

$4 + 6 z = - 14$

$y = - z$

$x = 4 - y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 4 - y - z$ with $- z$ .

$x = 4 - (- z) - z$

$4 + 6 z = - 14$

$y = - z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $4 - (- z) - z$ .

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Step 4.4.1.1

Multiply $- (- z)$ .

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Step 4.4.1.1.1

Multiply $- 1$ by $- 1$ .

$x = 4 + 1 z - z$

$4 + 6 z = - 14$

$y = - z$

Step 4.4.1.1.2

Multiply $z$ by $1$ .

$x = 4 + z - z$

$4 + 6 z = - 14$

$y = - z$

$x = 4 + z - z$

$4 + 6 z = - 14$

$y = - z$

Step 4.4.1.2

Combine the opposite terms in $4 + z - z$ .

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Step 4.4.1.2.1

Subtract $z$ from $z$ .

$x = 4 + 0$

$4 + 6 z = - 14$

$y = - z$

Step 4.4.1.2.2

Add $4$ and $0$ .

$x = 4$

$4 + 6 z = - 14$

$y = - z$

$x = 4$

$4 + 6 z = - 14$

$y = - z$

$x = 4$

$4 + 6 z = - 14$

$y = - z$

$x = 4$

$4 + 6 z = - 14$

$y = - z$

$x = 4$

$4 + 6 z = - 14$

$y = - z$

Step 5

Solve for $z$ in $4 + 6 z = - 14$ .

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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 $4$ from both sides of the equation.

$6 z = - 14 - 4$

$x = 4$

$y = - z$

Step 5.1.2

Subtract $4$ from $- 14$ .

$6 z = - 18$

$x = 4$

$y = - z$

$6 z = - 18$

$x = 4$

$y = - z$

Step 5.2

Divide each term in $6 z = - 18$ by $6$ and simplify.

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Step 5.2.1

Divide each term in $6 z = - 18$ by $6$ .

$\frac{6 z}{6} = \frac{- 18}{6}$

$x = 4$

$y = - z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $6$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{6 z}{6} = \frac{- 18}{6}$

$x = 4$

$y = - z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 18}{6}$

$x = 4$

$y = - z$

$z = \frac{- 18}{6}$

$x = 4$

$y = - z$

$z = \frac{- 18}{6}$

$x = 4$

$y = - z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 18$ by $6$ .

$z = - 3$

$x = 4$

$y = - z$

$z = - 3$

$x = 4$

$y = - z$

$z = - 3$

$x = 4$

$y = - z$

$z = - 3$

$x = 4$

$y = - z$

Step 6

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

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Step 6.1

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

$y = - (- 3)$

$z = - 3$

$x = 4$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $- 1$ by $- 3$ .

$y = 3$

$z = - 3$

$x = 4$

$y = 3$

$z = - 3$

$x = 4$

$y = 3$

$z = - 3$

$x = 4$

Step 7

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

$(4, 3, - 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(4, 3, - 3)$

Equation Form:

$x = 4, y = 3, z = - 3$