Finite Math Examples

$2 x y + 1 = 0$ , $x + 36 y = 3$

Step 1

Subtract $36 y$ from both sides of the equation.

$x = 3 - 36 y$

$2 x y + 1 = 0$

Step 2

Replace all occurrences of $x$ with $3 - 36 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x y + 1 = 0$ with $3 - 36 y$ .

$2 (3 - 36 y) \cdot y + 1 = 0$

$x = 3 - 36 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$(2 \cdot 3 + 2 (- 36 y)) \cdot y + 1 = 0$

$x = 3 - 36 y$

Step 2.2.1.2

Multiply $2$ by $3$ .

$(6 + 2 (- 36 y)) \cdot y + 1 = 0$

$x = 3 - 36 y$

Step 2.2.1.3

Multiply $- 36$ by $2$ .

$(6 - 72 y) \cdot y + 1 = 0$

$x = 3 - 36 y$

Step 2.2.1.4

Apply the distributive property.

$6 y - 72 y \cdot y + 1 = 0$

$x = 3 - 36 y$

Step 2.2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.1.5.1

Move $y$ .

$6 y - 72 (y \cdot y) + 1 = 0$

$x = 3 - 36 y$

Step 2.2.1.5.2

Multiply $y$ by $y$ .

$6 y - 72 y^{2} + 1 = 0$

$x = 3 - 36 y$

$6 y - 72 y^{2} + 1 = 0$

$x = 3 - 36 y$

$6 y - 72 y^{2} + 1 = 0$

$x = 3 - 36 y$

$6 y - 72 y^{2} + 1 = 0$

$x = 3 - 36 y$

$6 y - 72 y^{2} + 1 = 0$

$x = 3 - 36 y$

Step 3

Solve for $y$ in $6 y - 72 y^{2} + 1 = 0$ .

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Step 3.1

Factor the left side of the equation.

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Step 3.1.1

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$6 u - 72 u^{2} + 1 = 0$

$x = 3 - 36 y$

Step 3.1.2

Factor by grouping.

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Step 3.1.2.1

Reorder terms.

$- 72 u^{2} + 6 u + 1 = 0$

$x = 3 - 36 y$

Step 3.1.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 72 \cdot 1 = - 72$ and whose sum is $b = 6$ .

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Step 3.1.2.2.1

Factor $6$ out of $6 u$ .

$- 72 u^{2} + 6 (u) + 1 = 0$

$x = 3 - 36 y$

Step 3.1.2.2.2

Rewrite $6$ as $- 6$ plus $12$

$- 72 u^{2} + (- 6 + 12) u + 1 = 0$

$x = 3 - 36 y$

Step 3.1.2.2.3

Apply the distributive property.

$- 72 u^{2} - 6 u + 12 u + 1 = 0$

$x = 3 - 36 y$

$- 72 u^{2} - 6 u + 12 u + 1 = 0$

$x = 3 - 36 y$

Step 3.1.2.3

Factor out the greatest common factor from each group.

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Step 3.1.2.3.1

Group the first two terms and the last two terms.

$(- 72 u^{2} - 6 u) + 12 u + 1 = 0$

$x = 3 - 36 y$

Step 3.1.2.3.2

Factor out the greatest common factor (GCF) from each group.

$6 u (- 12 u - 1) - (- 12 u - 1) = 0$

$x = 3 - 36 y$

$6 u (- 12 u - 1) - (- 12 u - 1) = 0$

$x = 3 - 36 y$

Step 3.1.2.4

Factor the polynomial by factoring out the greatest common factor, $- 12 u - 1$ .

$(- 12 u - 1) (6 u - 1) = 0$

$x = 3 - 36 y$

$(- 12 u - 1) (6 u - 1) = 0$

$x = 3 - 36 y$

Step 3.1.3

Replace all occurrences of $u$ with $y$ .

$(- 12 y - 1) (6 y - 1) = 0$

$x = 3 - 36 y$

$(- 12 y - 1) (6 y - 1) = 0$

$x = 3 - 36 y$

Step 3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$- 12 y - 1 = 0$

$6 y - 1 = 0$

$x = 3 - 36 y$

Step 3.3

Set $- 12 y - 1$ equal to $0$ and solve for $y$ .

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Step 3.3.1

Set $- 12 y - 1$ equal to $0$ .

$- 12 y - 1 = 0$

$x = 3 - 36 y$

Step 3.3.2

Solve $- 12 y - 1 = 0$ for $y$ .

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Step 3.3.2.1

Add $1$ to both sides of the equation.

$- 12 y = 1$

$x = 3 - 36 y$

Step 3.3.2.2

Divide each term in $- 12 y = 1$ by $- 12$ and simplify.

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Step 3.3.2.2.1

Divide each term in $- 12 y = 1$ by $- 12$ .

$\frac{- 12 y}{- 12} = \frac{1}{- 12}$

$x = 3 - 36 y$

Step 3.3.2.2.2

Simplify the left side.

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Step 3.3.2.2.2.1

Cancel the common factor of $- 12$ .

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Step 3.3.2.2.2.1.1

Cancel the common factor.

$\frac{- 12 y}{- 12} = \frac{1}{- 12}$

$x = 3 - 36 y$

Step 3.3.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{- 12}$

$x = 3 - 36 y$

$y = \frac{1}{- 12}$

$x = 3 - 36 y$

$y = \frac{1}{- 12}$

$x = 3 - 36 y$

Step 3.3.2.2.3

Simplify the right side.

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Step 3.3.2.2.3.1

Move the negative in front of the fraction.

$y = - \frac{1}{12}$

$x = 3 - 36 y$

$y = - \frac{1}{12}$

$x = 3 - 36 y$

$y = - \frac{1}{12}$

$x = 3 - 36 y$

$y = - \frac{1}{12}$

$x = 3 - 36 y$

$y = - \frac{1}{12}$

$x = 3 - 36 y$

Step 3.4

Set $6 y - 1$ equal to $0$ and solve for $y$ .

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Step 3.4.1

Set $6 y - 1$ equal to $0$ .

$6 y - 1 = 0$

$x = 3 - 36 y$

Step 3.4.2

Solve $6 y - 1 = 0$ for $y$ .

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Step 3.4.2.1

Add $1$ to both sides of the equation.

$6 y = 1$

$x = 3 - 36 y$

Step 3.4.2.2

Divide each term in $6 y = 1$ by $6$ and simplify.

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Step 3.4.2.2.1

Divide each term in $6 y = 1$ by $6$ .

$\frac{6 y}{6} = \frac{1}{6}$

$x = 3 - 36 y$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $6$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{1}{6}$

$x = 3 - 36 y$

Step 3.4.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{6}$

$x = 3 - 36 y$

$y = \frac{1}{6}$

$x = 3 - 36 y$

$y = \frac{1}{6}$

$x = 3 - 36 y$

$y = \frac{1}{6}$

$x = 3 - 36 y$

$y = \frac{1}{6}$

$x = 3 - 36 y$

$y = \frac{1}{6}$

$x = 3 - 36 y$

Step 3.5

The final solution is all the values that make $(- 12 y - 1) (6 y - 1) = 0$ true.

$y = - \frac{1}{12}, \frac{1}{6}$

$x = 3 - 36 y$

$y = - \frac{1}{12}, \frac{1}{6}$

$x = 3 - 36 y$

Step 4

Replace all occurrences of $y$ with $- \frac{1}{12}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 3 - 36 y$ with $- \frac{1}{12}$ .

$x = 3 - 36 (- \frac{1}{12})$

$y = - \frac{1}{12}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $3 - 36 (- \frac{1}{12})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $12$ .

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Step 4.2.1.1.1.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$x = 3 - 36 (\frac{- 1}{12})$

$y = - \frac{1}{12}$

Step 4.2.1.1.1.2

Factor $12$ out of $- 36$ .

$x = 3 + 12 (- 3) (\frac{- 1}{12})$

$y = - \frac{1}{12}$

Step 4.2.1.1.1.3

Cancel the common factor.

$x = 3 + 12 \cdot (- 3 (\frac{- 1}{12}))$

$y = - \frac{1}{12}$

Step 4.2.1.1.1.4

Rewrite the expression.

$x = 3 - 3 \cdot - 1$

$y = - \frac{1}{12}$

$x = 3 - 3 \cdot - 1$

$y = - \frac{1}{12}$

Step 4.2.1.1.2

Multiply $- 3$ by $- 1$ .

$x = 3 + 3$

$y = - \frac{1}{12}$

$x = 3 + 3$

$y = - \frac{1}{12}$

Step 4.2.1.2

Add $3$ and $3$ .

$x = 6$

$y = - \frac{1}{12}$

$x = 6$

$y = - \frac{1}{12}$

$x = 6$

$y = - \frac{1}{12}$

$x = 6$

$y = - \frac{1}{12}$

Step 5

Replace all occurrences of $y$ with $\frac{1}{6}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 3 - 36 y$ with $\frac{1}{6}$ .

$x = 3 - 36 (\frac{1}{6})$

$y = \frac{1}{6}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $3 - 36 (\frac{1}{6})$ .

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Step 5.2.1.1

Cancel the common factor of $6$ .

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Step 5.2.1.1.1

Factor $6$ out of $- 36$ .

$x = 3 + 6 (- 6) (\frac{1}{6})$

$y = \frac{1}{6}$

Step 5.2.1.1.2

Cancel the common factor.

$x = 3 + 6 \cdot (- 6 (\frac{1}{6}))$

$y = \frac{1}{6}$

Step 5.2.1.1.3

Rewrite the expression.

$x = 3 - 6$

$y = \frac{1}{6}$

$x = 3 - 6$

$y = \frac{1}{6}$

Step 5.2.1.2

Subtract $6$ from $3$ .

$x = - 3$

$y = \frac{1}{6}$

$x = - 3$

$y = \frac{1}{6}$

$x = - 3$

$y = \frac{1}{6}$

$x = - 3$

$y = \frac{1}{6}$

Step 6

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

$(6, - \frac{1}{12})$

$(- 3, \frac{1}{6})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(6, - \frac{1}{12}), (- 3, \frac{1}{6})$

Equation Form:

$x = 6, y = - \frac{1}{12}$

$x = - 3, y = \frac{1}{6}$

Step 8