Finite Math Examples

$x + y + z = - 3$ , $x - y + 4 z = 18$ , $5 x + y + z = - 23$

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.

$x + z = - 3 - y$

$x - y + 4 z = 18$

$5 x + y + z = - 23$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = - 3 - y - z$

$x - y + 4 z = 18$

$5 x + y + z = - 23$

$x = - 3 - y - z$

$x - y + 4 z = 18$

$5 x + y + z = - 23$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x - y + 4 z = 18$ with $- 3 - y - z$ .

$(- 3 - y - z) - y + 4 z = 18$

$x = - 3 - y - z$

$5 x + y + z = - 23$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $(- 3 - y - z) - y + 4 z$ .

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Step 2.2.1.1

Subtract $y$ from $- y$ .

$- 3 - 2 y - z + 4 z = 18$

$x = - 3 - y - z$

$5 x + y + z = - 23$

Step 2.2.1.2

Add $- z$ and $4 z$ .

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$5 x + y + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$5 x + y + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$5 x + y + z = - 23$

Step 2.3

Replace all occurrences of $x$ in $5 x + y + z = - 23$ with $- 3 - y - z$ .

$5 (- 3 - y - z) + y + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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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.

$5 \cdot - 3 + 5 (- y) + 5 (- z) + y + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $5$ by $- 3$ .

$- 15 + 5 (- y) + 5 (- z) + y + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $5$ .

$- 15 - 5 y + 5 (- z) + y + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $5$ .

$- 15 - 5 y - 5 z + y + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$- 15 - 5 y - 5 z + y + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$- 15 - 5 y - 5 z + y + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 5 y$ and $y$ .

$- 15 - 4 y - 5 z + z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 2.4.1.2.2

Add $- 5 z$ and $z$ .

$- 15 - 4 y - 4 z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$- 15 - 4 y - 4 z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$- 15 - 4 y - 4 z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$- 15 - 4 y - 4 z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$- 15 - 4 y - 4 z = - 23$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 3

Solve for $y$ in $- 15 - 4 y - 4 z = - 23$ .

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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

Add $15$ to both sides of the equation.

$- 4 y - 4 z = - 23 + 15$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 3.1.2

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

$- 4 y = - 23 + 15 + 4 z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 3.1.3

Add $- 23$ and $15$ .

$- 4 y = - 8 + 4 z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$- 4 y = - 8 + 4 z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 3.2

Divide each term in $- 4 y = - 8 + 4 z$ by $- 4$ and simplify.

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Step 3.2.1

Divide each term in $- 4 y = - 8 + 4 z$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{- 8}{- 4} + \frac{4 z}{- 4}$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

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.

$\frac{- 4 y}{- 4} = \frac{- 8}{- 4} + \frac{4 z}{- 4}$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 8}{- 4} + \frac{4 z}{- 4}$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$y = \frac{- 8}{- 4} + \frac{4 z}{- 4}$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$y = \frac{- 8}{- 4} + \frac{4 z}{- 4}$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $- 8$ by $- 4$ .

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

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 3.2.3.1.2

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

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Step 3.2.3.1.2.1

Factor $4$ out of $4 z$ .

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

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 3.2.3.1.2.2

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

$y = 2 - 1 \cdot z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$y = 2 - 1 \cdot z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 3.2.3.1.3

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

$y = 2 - z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$y = 2 - z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$y = 2 - z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$y = 2 - z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

$y = 2 - z$

$- 3 - 2 y + 3 z = 18$

$x = - 3 - y - z$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $- 3 - 2 y + 3 z = 18$ with $2 - z$ .

$- 3 - 2 (2 - z) + 3 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 3 - 2 \cdot 2 - 2 (- z) + 3 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

Step 4.2.1.1.2

Multiply $- 2$ by $2$ .

$- 3 - 4 - 2 (- z) + 3 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

Step 4.2.1.1.3

Multiply $- 1$ by $- 2$ .

$- 3 - 4 + 2 z + 3 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

$- 3 - 4 + 2 z + 3 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $4$ from $- 3$ .

$- 7 + 2 z + 3 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

Step 4.2.1.2.2

Add $2 z$ and $3 z$ .

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 3 - y - z$

Step 4.3

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

$x = - 3 - (2 - z) - z$

$- 7 + 5 z = 18$

$y = 2 - z$

Step 4.4

Simplify the right side.

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Step 4.4.1

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

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$x = - 3 - 1 \cdot 2 + z - z$

$- 7 + 5 z = 18$

$y = 2 - z$

Step 4.4.1.1.2

Multiply $- 1$ by $2$ .

$x = - 3 - 2 + z - z$

$- 7 + 5 z = 18$

$y = 2 - z$

Step 4.4.1.1.3

Multiply $- - z$ .

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Step 4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$x = - 3 - 2 + 1 z - z$

$- 7 + 5 z = 18$

$y = 2 - z$

Step 4.4.1.1.3.2

Multiply $z$ by $1$ .

$x = - 3 - 2 + z - z$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 3 - 2 + z - z$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 3 - 2 + z - z$

$- 7 + 5 z = 18$

$y = 2 - z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

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

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Step 4.4.1.2.1.1

Subtract $z$ from $z$ .

$x = - 3 - 2 + 0$

$- 7 + 5 z = 18$

$y = 2 - z$

Step 4.4.1.2.1.2

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

$x = - 3 - 2$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 3 - 2$

$- 7 + 5 z = 18$

$y = 2 - z$

Step 4.4.1.2.2

Subtract $2$ from $- 3$ .

$x = - 5$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 5$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 5$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 5$

$- 7 + 5 z = 18$

$y = 2 - z$

$x = - 5$

$- 7 + 5 z = 18$

$y = 2 - z$

Step 5

Solve for $z$ in $- 7 + 5 z = 18$ .

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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

Add $7$ to both sides of the equation.

$5 z = 18 + 7$

$x = - 5$

$y = 2 - z$

Step 5.1.2

Add $18$ and $7$ .

$5 z = 25$

$x = - 5$

$y = 2 - z$

$5 z = 25$

$x = - 5$

$y = 2 - z$

Step 5.2

Divide each term in $5 z = 25$ by $5$ and simplify.

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Step 5.2.1

Divide each term in $5 z = 25$ by $5$ .

$\frac{5 z}{5} = \frac{25}{5}$

$x = - 5$

$y = 2 - z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{5 z}{5} = \frac{25}{5}$

$x = - 5$

$y = 2 - z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{25}{5}$

$x = - 5$

$y = 2 - z$

$z = \frac{25}{5}$

$x = - 5$

$y = 2 - z$

$z = \frac{25}{5}$

$x = - 5$

$y = 2 - z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $25$ by $5$ .

$z = 5$

$x = - 5$

$y = 2 - z$

$z = 5$

$x = - 5$

$y = 2 - z$

$z = 5$

$x = - 5$

$y = 2 - z$

$z = 5$

$x = - 5$

$y = 2 - z$

Step 6

Replace all occurrences of $z$ with $5$ in each equation.

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Step 6.1

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

$y = 2 - (5)$

$z = 5$

$x = - 5$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $2 - (5)$ .

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Step 6.2.1.1

Multiply $- 1$ by $5$ .

$y = 2 - 5$

$z = 5$

$x = - 5$

Step 6.2.1.2

Subtract $5$ from $2$ .

$y = - 3$

$z = 5$

$x = - 5$

$y = - 3$

$z = 5$

$x = - 5$

$y = - 3$

$z = 5$

$x = - 5$

$y = - 3$

$z = 5$

$x = - 5$

Step 7

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

$(- 5, - 3, 5)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 5, - 3, 5)$

Equation Form:

$x = - 5, y = - 3, z = 5$