Finite Math Examples

$13 = 3 x - y$ , $4 y - 3 x + 2 z = - 3$ , $z = 2 x - 4 y$

Step 1

Replace all occurrences of $z$ with $2 x - 4 y$ in each equation.

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Step 1.1

Replace all occurrences of $z$ in $4 y - 3 x + 2 z = - 3$ with $2 x - 4 y$ .

$4 y - 3 x + 2 (2 x - 4 y) = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $4 y - 3 x + 2 (2 x - 4 y)$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Apply the distributive property.

$4 y - 3 x + 2 (2 x) + 2 (- 4 y) = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

Step 1.2.1.1.2

Multiply $2$ by $2$ .

$4 y - 3 x + 4 x + 2 (- 4 y) = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

Step 1.2.1.1.3

Multiply $- 4$ by $2$ .

$4 y - 3 x + 4 x - 8 y = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

$4 y - 3 x + 4 x - 8 y = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

Step 1.2.1.2

Simplify by adding terms.

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Step 1.2.1.2.1

Subtract $8 y$ from $4 y$ .

$- 3 x + 4 x - 4 y = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

Step 1.2.1.2.2

Add $- 3 x$ and $4 x$ .

$x - 4 y = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

$x - 4 y = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

$x - 4 y = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

$x - 4 y = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

$x - 4 y = - 3$

$13 = 3 x - y$

$z = 2 x - 4 y$

Step 2

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

$x = - 3 + 4 y$

$13 = 3 x - y$

$z = 2 x - 4 y$

Step 3

Replace all occurrences of $x$ with $- 3 + 4 y$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $13 = 3 x - y$ with $- 3 + 4 y$ .

$13 = 3 (- 3 + 4 y) - y$

$x = - 3 + 4 y$

$z = 2 x - 4 y$

Step 3.2

Simplify the right side.

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Step 3.2.1

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

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Apply the distributive property.

$13 = 3 \cdot - 3 + 3 (4 y) - y$

$x = - 3 + 4 y$

$z = 2 x - 4 y$

Step 3.2.1.1.2

Multiply $3$ by $- 3$ .

$13 = - 9 + 3 (4 y) - y$

$x = - 3 + 4 y$

$z = 2 x - 4 y$

Step 3.2.1.1.3

Multiply $4$ by $3$ .

$13 = - 9 + 12 y - y$

$x = - 3 + 4 y$

$z = 2 x - 4 y$

$13 = - 9 + 12 y - y$

$x = - 3 + 4 y$

$z = 2 x - 4 y$

Step 3.2.1.2

Subtract $y$ from $12 y$ .

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

$z = 2 x - 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

$z = 2 x - 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

$z = 2 x - 4 y$

Step 3.3

Replace all occurrences of $x$ in $z = 2 x - 4 y$ with $- 3 + 4 y$ .

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

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

Step 3.4

Simplify the right side.

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Step 3.4.1

Simplify $2 (- 3 + 4 y) - 4 y$ .

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Step 3.4.1.1

Simplify each term.

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Step 3.4.1.1.1

Apply the distributive property.

$z = 2 \cdot - 3 + 2 (4 y) - 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

Step 3.4.1.1.2

Multiply $2$ by $- 3$ .

$z = - 6 + 2 (4 y) - 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

Step 3.4.1.1.3

Multiply $4$ by $2$ .

$z = - 6 + 8 y - 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

$z = - 6 + 8 y - 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

Step 3.4.1.2

Subtract $4 y$ from $8 y$ .

$z = - 6 + 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

$z = - 6 + 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

$z = - 6 + 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

$z = - 6 + 4 y$

$13 = - 9 + 11 y$

$x = - 3 + 4 y$

Step 4

Solve for $y$ in $13 = - 9 + 11 y$ .

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Step 4.1

Rewrite the equation as $- 9 + 11 y = 13$ .

$- 9 + 11 y = 13$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

Step 4.2

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

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Step 4.2.1

Add $9$ to both sides of the equation.

$11 y = 13 + 9$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

Step 4.2.2

Add $13$ and $9$ .

$11 y = 22$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

$11 y = 22$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

Step 4.3

Divide each term in $11 y = 22$ by $11$ and simplify.

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Step 4.3.1

Divide each term in $11 y = 22$ by $11$ .

$\frac{11 y}{11} = \frac{22}{11}$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $11$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{11 y}{11} = \frac{22}{11}$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

Step 4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{22}{11}$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

$y = \frac{22}{11}$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

$y = \frac{22}{11}$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $22$ by $11$ .

$y = 2$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

$y = 2$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

$y = 2$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

$y = 2$

$z = - 6 + 4 y$

$x = - 3 + 4 y$

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 $z = - 6 + 4 y$ with $2$ .

$z = - 6 + 4 (2)$

$y = 2$

$x = - 3 + 4 y$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 6 + 4 (2)$ .

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Step 5.2.1.1

Multiply $4$ by $2$ .

$z = - 6 + 8$

$y = 2$

$x = - 3 + 4 y$

Step 5.2.1.2

Add $- 6$ and $8$ .

$z = 2$

$y = 2$

$x = - 3 + 4 y$

$z = 2$

$y = 2$

$x = - 3 + 4 y$

$z = 2$

$y = 2$

$x = - 3 + 4 y$

Step 5.3

Replace all occurrences of $y$ in $x = - 3 + 4 y$ with $2$ .

$x = - 3 + 4 (2)$

$z = 2$

$y = 2$

Step 5.4

Simplify the right side.

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Step 5.4.1

Simplify $- 3 + 4 (2)$ .

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Step 5.4.1.1

Multiply $4$ by $2$ .

$x = - 3 + 8$

$z = 2$

$y = 2$

Step 5.4.1.2

Add $- 3$ and $8$ .

$x = 5$

$z = 2$

$y = 2$

$x = 5$

$z = 2$

$y = 2$

$x = 5$

$z = 2$

$y = 2$

$x = 5$

$z = 2$

$y = 2$

Step 6

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

$(5, 2, 2)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(5, 2, 2)$

Equation Form:

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