Finite Math Examples

$4 x - y + 9 z = 92$ , $3 x + 3 y + 3 z = 54$ , $- 7 x - 7 y + z = - 62$

Step 1

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

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Step 1.1

Add $7 x$ to both sides of the equation.

$- 7 y + z = - 62 + 7 x$

$4 x - y + 9 z = 92$

$3 x + 3 y + 3 z = 54$

Step 1.2

Add $7 y$ to both sides of the equation.

$z = - 62 + 7 x + 7 y$

$4 x - y + 9 z = 92$

$3 x + 3 y + 3 z = 54$

$z = - 62 + 7 x + 7 y$

$4 x - y + 9 z = 92$

$3 x + 3 y + 3 z = 54$

Step 2

Replace all occurrences of $z$ with $- 62 + 7 x + 7 y$ in each equation.

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Step 2.1

Replace all occurrences of $z$ in $4 x - y + 9 z = 92$ with $- 62 + 7 x + 7 y$ .

$4 x - y + 9 (- 62 + 7 x + 7 y) = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 x - y + 9 (- 62 + 7 x + 7 y)$ .

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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.

$4 x - y + 9 \cdot - 62 + 9 (7 x) + 9 (7 y) = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $9$ by $- 62$ .

$4 x - y - 558 + 9 (7 x) + 9 (7 y) = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

Step 2.2.1.1.2.2

Multiply $7$ by $9$ .

$4 x - y - 558 + 63 x + 9 (7 y) = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

Step 2.2.1.1.2.3

Multiply $7$ by $9$ .

$4 x - y - 558 + 63 x + 63 y = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

$4 x - y - 558 + 63 x + 63 y = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

$4 x - y - 558 + 63 x + 63 y = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $4 x$ and $63 x$ .

$67 x - y - 558 + 63 y = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

Step 2.2.1.2.2

Add $- y$ and $63 y$ .

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y + 3 z = 54$

Step 2.3

Replace all occurrences of $z$ in $3 x + 3 y + 3 z = 54$ with $- 62 + 7 x + 7 y$ .

$3 x + 3 y + 3 (- 62 + 7 x + 7 y) = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $3 x + 3 y + 3 (- 62 + 7 x + 7 y)$ .

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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.

$3 x + 3 y + 3 \cdot - 62 + 3 (7 x) + 3 (7 y) = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $3$ by $- 62$ .

$3 x + 3 y - 186 + 3 (7 x) + 3 (7 y) = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 2.4.1.1.2.2

Multiply $7$ by $3$ .

$3 x + 3 y - 186 + 21 x + 3 (7 y) = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 2.4.1.1.2.3

Multiply $7$ by $3$ .

$3 x + 3 y - 186 + 21 x + 21 y = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y - 186 + 21 x + 21 y = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$3 x + 3 y - 186 + 21 x + 21 y = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $3 x$ and $21 x$ .

$24 x + 3 y - 186 + 21 y = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 2.4.1.2.2

Add $3 y$ and $21 y$ .

$24 x + 24 y - 186 = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$24 x + 24 y - 186 = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$24 x + 24 y - 186 = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$24 x + 24 y - 186 = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$24 x + 24 y - 186 = 54$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3

Solve for $x$ in $24 x + 24 y - 186 = 54$ .

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Step 3.1

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

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Step 3.1.1

Subtract $24 y$ from both sides of the equation.

$24 x - 186 = 54 - 24 y$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.1.2

Add $186$ to both sides of the equation.

$24 x = 54 - 24 y + 186$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.1.3

Add $54$ and $186$ .

$24 x = - 24 y + 240$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$24 x = - 24 y + 240$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.2

Divide each term in $24 x = - 24 y + 240$ by $24$ and simplify.

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Step 3.2.1

Divide each term in $24 x = - 24 y + 240$ by $24$ .

$\frac{24 x}{24} = \frac{- 24 y}{24} + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $24$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{24 x}{24} = \frac{- 24 y}{24} + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 24 y}{24} + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$x = \frac{- 24 y}{24} + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$x = \frac{- 24 y}{24} + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

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

Cancel the common factor of $- 24$ and $24$ .

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Step 3.2.3.1.1.1

Factor $24$ out of $- 24 y$ .

$x = \frac{24 (- y)}{24} + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $24$ out of $24$ .

$x = \frac{24 (- y)}{24 (1)} + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{24 (- y)}{24 \cdot 1} + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- y}{1} + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.2.3.1.1.2.4

Divide $- y$ by $1$ .

$x = - y + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$x = - y + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$x = - y + \frac{240}{24}$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 3.2.3.1.2

Divide $240$ by $24$ .

$x = - y + 10$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$x = - y + 10$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$x = - y + 10$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$x = - y + 10$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

$x = - y + 10$

$67 x + 62 y - 558 = 92$

$z = - 62 + 7 x + 7 y$

Step 4

Replace all occurrences of $x$ with $- y + 10$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $67 x + 62 y - 558 = 92$ with $- y + 10$ .

$67 (- y + 10) + 62 y - 558 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $67 (- y + 10) + 62 y - 558$ .

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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.

$67 (- y) + 67 \cdot 10 + 62 y - 558 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

Step 4.2.1.1.2

Multiply $- 1$ by $67$ .

$- 67 y + 67 \cdot 10 + 62 y - 558 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

Step 4.2.1.1.3

Multiply $67$ by $10$ .

$- 67 y + 670 + 62 y - 558 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

$- 67 y + 670 + 62 y - 558 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $- 67 y$ and $62 y$ .

$- 5 y + 670 - 558 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

Step 4.2.1.2.2

Subtract $558$ from $670$ .

$- 5 y + 112 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

$- 5 y + 112 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

$- 5 y + 112 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

$- 5 y + 112 = 92$

$x = - y + 10$

$z = - 62 + 7 x + 7 y$

Step 4.3

Replace all occurrences of $x$ in $z = - 62 + 7 x + 7 y$ with $- y + 10$ .

$z = - 62 + 7 (- y + 10) + 7 y$

$- 5 y + 112 = 92$

$x = - y + 10$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 62 + 7 (- y + 10) + 7 y$ .

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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.

$z = - 62 + 7 (- y) + 7 \cdot 10 + 7 y$

$- 5 y + 112 = 92$

$x = - y + 10$

Step 4.4.1.1.2

Multiply $- 1$ by $7$ .

$z = - 62 - 7 y + 7 \cdot 10 + 7 y$

$- 5 y + 112 = 92$

$x = - y + 10$

Step 4.4.1.1.3

Multiply $7$ by $10$ .

$z = - 62 - 7 y + 70 + 7 y$

$- 5 y + 112 = 92$

$x = - y + 10$

$z = - 62 - 7 y + 70 + 7 y$

$- 5 y + 112 = 92$

$x = - y + 10$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Combine the opposite terms in $- 62 - 7 y + 70 + 7 y$ .

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Step 4.4.1.2.1.1

Add $- 7 y$ and $7 y$ .

$z = - 62 + 0 + 70$

$- 5 y + 112 = 92$

$x = - y + 10$

Step 4.4.1.2.1.2

Add $- 62$ and $0$ .

$z = - 62 + 70$

$- 5 y + 112 = 92$

$x = - y + 10$

$z = - 62 + 70$

$- 5 y + 112 = 92$

$x = - y + 10$

Step 4.4.1.2.2

Add $- 62$ and $70$ .

$z = 8$

$- 5 y + 112 = 92$

$x = - y + 10$

$z = 8$

$- 5 y + 112 = 92$

$x = - y + 10$

$z = 8$

$- 5 y + 112 = 92$

$x = - y + 10$

$z = 8$

$- 5 y + 112 = 92$

$x = - y + 10$

$z = 8$

$- 5 y + 112 = 92$

$x = - y + 10$

Step 5

Solve for $y$ in $- 5 y + 112 = 92$ .

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Step 5.1

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

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Step 5.1.1

Subtract $112$ from both sides of the equation.

$- 5 y = 92 - 112$

$z = 8$

$x = - y + 10$

Step 5.1.2

Subtract $112$ from $92$ .

$- 5 y = - 20$

$z = 8$

$x = - y + 10$

$- 5 y = - 20$

$z = 8$

$x = - y + 10$

Step 5.2

Divide each term in $- 5 y = - 20$ by $- 5$ and simplify.

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Step 5.2.1

Divide each term in $- 5 y = - 20$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 20}{- 5}$

$z = 8$

$x = - y + 10$

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.

$\frac{- 5 y}{- 5} = \frac{- 20}{- 5}$

$z = 8$

$x = - y + 10$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 20}{- 5}$

$z = 8$

$x = - y + 10$

$y = \frac{- 20}{- 5}$

$z = 8$

$x = - y + 10$

$y = \frac{- 20}{- 5}$

$z = 8$

$x = - y + 10$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 20$ by $- 5$ .

$y = 4$

$z = 8$

$x = - y + 10$

$y = 4$

$z = 8$

$x = - y + 10$

$y = 4$

$z = 8$

$x = - y + 10$

$y = 4$

$z = 8$

$x = - y + 10$

Step 6

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

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Step 6.1

Replace all occurrences of $y$ in $x = - y + 10$ with $4$ .

$x = - (4) + 10$

$y = 4$

$z = 8$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- (4) + 10$ .

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Step 6.2.1.1

Multiply $- 1$ by $4$ .

$x = - 4 + 10$

$y = 4$

$z = 8$

Step 6.2.1.2

Add $- 4$ and $10$ .

$x = 6$

$y = 4$

$z = 8$

$x = 6$

$y = 4$

$z = 8$

$x = 6$

$y = 4$

$z = 8$

$x = 6$

$y = 4$

$z = 8$

Step 7

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

$(6, 4, 8)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(6, 4, 8)$

Equation Form:

$x = 6, y = 4, z = 8$