Finite Math Examples

$\frac{9}{2 x + 6} = 2 y (x + 3)$

Step 1

Subtract $2 y (x + 3)$ from both sides of the equation.

$\frac{9}{2 x + 6} - 2 y (x + 3) = 0$

Step 2

Simplify each term.

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Step 2.1

Factor $2$ out of $2 x + 6$ .

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Step 2.1.1

Factor $2$ out of $2 x$ .

$\frac{9}{2 (x) + 6} - 2 y (x + 3) = 0$

Step 2.1.2

Factor $2$ out of $6$ .

$\frac{9}{2 x + 2 \cdot 3} - 2 y (x + 3) = 0$

Step 2.1.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

$\frac{9}{2 (x + 3)} - 2 y (x + 3) = 0$

Step 2.2

Apply the distributive property.

$\frac{9}{2 (x + 3)} - 2 y x - 2 y \cdot 3 = 0$

Step 2.3

Multiply $3$ by $- 2$ .

$\frac{9}{2 (x + 3)} - 2 y x - 6 y = 0$

Step 3

To write $- 2 y x$ as a fraction with a common denominator, multiply by $\frac{2 (x + 3)}{2 (x + 3)}$ .

$\frac{9}{2 (x + 3)} - 2 y x \cdot \frac{2 (x + 3)}{2 (x + 3)} - 6 y = 0$

Step 4

Combine $- 2 y x$ and $\frac{2 (x + 3)}{2 (x + 3)}$ .

$\frac{9}{2 (x + 3)} + \frac{- 2 y x (2 (x + 3))}{2 (x + 3)} - 6 y = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{9 - 2 y x (2 (x + 3))}{2 (x + 3)} - 6 y = 0$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{9 - 2 y x (2 x + 2 \cdot 3)}{2 (x + 3)} - 6 y = 0$

Step 6.2

Multiply $2$ by $3$ .

$\frac{9 - 2 y x (2 x + 6)}{2 (x + 3)} - 6 y = 0$

Step 6.3

Apply the distributive property.

$\frac{9 - 2 y x (2 x) - 2 y x \cdot 6}{2 (x + 3)} - 6 y = 0$

Step 6.4

Multiply $x$ by $x$ by adding the exponents.

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Step 6.4.1

Move $x$ .

$\frac{9 - 2 y (x \cdot x) \cdot 2 - 2 y x \cdot 6}{2 (x + 3)} - 6 y = 0$

Step 6.4.2

Multiply $x$ by $x$ .

$\frac{9 - 2 y x^{2} \cdot 2 - 2 y x \cdot 6}{2 (x + 3)} - 6 y = 0$

Step 6.5

Multiply $6$ by $- 2$ .

$\frac{9 - 2 y x^{2} \cdot 2 - 12 y x}{2 (x + 3)} - 6 y = 0$

Step 6.6

Multiply $2$ by $- 2$ .

$\frac{9 - 4 y x^{2} - 12 y x}{2 (x + 3)} - 6 y = 0$

Step 7

To write $- 6 y$ as a fraction with a common denominator, multiply by $\frac{2 (x + 3)}{2 (x + 3)}$ .

$\frac{9 - 4 y x^{2} - 12 y x}{2 (x + 3)} - 6 y \cdot \frac{2 (x + 3)}{2 (x + 3)} = 0$

Step 8

Combine $- 6 y$ and $\frac{2 (x + 3)}{2 (x + 3)}$ .

$\frac{9 - 4 y x^{2} - 12 y x}{2 (x + 3)} + \frac{- 6 y (2 (x + 3))}{2 (x + 3)} = 0$

Step 9

Combine the numerators over the common denominator.

$\frac{9 - 4 y x^{2} - 12 y x - 6 y (2 (x + 3))}{2 (x + 3)} = 0$

Step 10

Remove parentheses.

$\frac{9 - 4 y x^{2} - 12 y x - 6 y \cdot (2 (x + 3))}{2 (x + 3)} = 0$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $2$ by $- 6$ .

$\frac{9 - 4 y x^{2} - 12 y x - 12 y (x + 3)}{2 (x + 3)} = 0$

Step 11.2

Apply the distributive property.

$\frac{9 - 4 y x^{2} - 12 y x - 12 y x - 12 y \cdot 3}{2 (x + 3)} = 0$

Step 11.3

Multiply $3$ by $- 12$ .

$\frac{9 - 4 y x^{2} - 12 y x - 12 y x - 36 y}{2 (x + 3)} = 0$

Step 11.4

Subtract $12 y x$ from $- 12 y x$ .

$\frac{9 - 4 y x^{2} - 24 y x - 36 y}{2 (x + 3)} = 0$

Step 12

Set the numerator equal to zero.

$9 - 4 y x^{2} - 24 y x - 36 y = 0$

Step 13

Solve the equation for $y$ .

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Step 13.1

Subtract $9$ from both sides of the equation.

$- 4 y x^{2} - 24 y x - 36 y = - 9$

Step 13.2

Factor $4 y$ out of $- 4 y x^{2} - 24 y x - 36 y$ .

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Step 13.2.1

Factor $4 y$ out of $- 4 y x^{2}$ .

$4 y (- 1 x^{2}) - 24 y x - 36 y = - 9$

Step 13.2.2

Factor $4 y$ out of $- 24 y x$ .

$4 y (- 1 x^{2}) + 4 y (- 6 x) - 36 y = - 9$

Step 13.2.3

Factor $4 y$ out of $- 36 y$ .

$4 y (- 1 x^{2}) + 4 y (- 6 x) + 4 y (- 9) = - 9$

Step 13.2.4

Factor $4 y$ out of $4 y (- 1 x^{2}) + 4 y (- 6 x)$ .

$4 y (- 1 x^{2} - 6 x) + 4 y (- 9) = - 9$

Step 13.2.5

Factor $4 y$ out of $4 y (- 1 x^{2} - 6 x) + 4 y (- 9)$ .

$4 y (- 1 x^{2} - 6 x - 9) = - 9$

Step 13.3

Factor by grouping.

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Step 13.3.1

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 $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = - 6$ .

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Step 13.3.1.1

Factor $- 6$ out of $- 6 x$ .

$4 y (- 1 x^{2} - 6 x - 9) = - 9$

Step 13.3.1.2

Rewrite $- 6$ as $- 3$ plus $- 3$

$4 y (- 1 x^{2} + (- 3 - 3) x - 9) = - 9$

Step 13.3.1.3

Apply the distributive property.

$4 y (- 1 x^{2} - 3 x - 3 x - 9) = - 9$

Step 13.3.2

Factor out the greatest common factor from each group.

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Step 13.3.2.1

Group the first two terms and the last two terms.

$4 y ((- 1 x^{2} - 3 x) - 3 x - 9) = - 9$

Step 13.3.2.2

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

$4 y (x (- 1 x - 3) + 3 (- x - 3)) = - 9$

Step 13.3.3

Factor the polynomial by factoring out the greatest common factor, $- 1 x - 3$ .

$4 y ((- 1 x - 3) (x + 3)) = - 9$

Step 13.4

Factor.

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Step 13.4.1

Rewrite $- 1 x$ as $- x$ .

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

Step 13.4.2

Remove unnecessary parentheses.

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

Step 13.5

Combine exponents.

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Step 13.5.1

Factor $- 1$ out of $- x$ .

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

Step 13.5.2

Rewrite $- 3$ as $- 1 (3)$ .

$4 y (- (x) - 1 \cdot 3) (x + 3) = - 9$

Step 13.5.3

Factor $- 1$ out of $- (x) - 1 (3)$ .

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

Step 13.5.4

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$4 y (- 1 (x + 3)) (x + 3) = - 9$

Step 13.5.5

Remove parentheses.

$4 y \cdot (- 1 (x + 3) (x + 3)) = - 9$

Step 13.5.6

Raise $x + 3$ to the power of $1$ .

$4 y \cdot (- 1 ((x + 3) (x + 3))) = - 9$

Step 13.5.7

Raise $x + 3$ to the power of $1$ .

$4 y \cdot (- 1 ((x + 3) (x + 3))) = - 9$

Step 13.5.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 y \cdot (- 1 {(x + 3)}^{1 + 1}) = - 9$

Step 13.5.9

Add $1$ and $1$ .

$4 y \cdot (- 1 {(x + 3)}^{2}) = - 9$

Step 13.5.10

Multiply $- 1$ by $4$ .

$- 4 y {(x + 3)}^{2} = - 9$

Step 13.6

Divide each term in $- 4 y {(x + 3)}^{2} = - 9$ by $- 4 {(x + 3)}^{2}$ and simplify.

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Step 13.6.1

Divide each term in $- 4 y {(x + 3)}^{2} = - 9$ by $- 4 {(x + 3)}^{2}$ .

$\frac{- 4 y {(x + 3)}^{2}}{- 4 {(x + 3)}^{2}} = \frac{- 9}{- 4 {(x + 3)}^{2}}$

Step 13.6.2

Simplify the left side.

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Step 13.6.2.1

Cancel the common factor of $- 4$ .

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Step 13.6.2.1.1

Cancel the common factor.

$\frac{- 4 y {(x + 3)}^{2}}{- 4 {(x + 3)}^{2}} = \frac{- 9}{- 4 {(x + 3)}^{2}}$

Step 13.6.2.1.2

Rewrite the expression.

$\frac{y {(x + 3)}^{2}}{{(x + 3)}^{2}} = \frac{- 9}{- 4 {(x + 3)}^{2}}$

Step 13.6.2.2

Cancel the common factor of ${(x + 3)}^{2}$ .

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Step 13.6.2.2.1

Cancel the common factor.

$\frac{y {(x + 3)}^{2}}{{(x + 3)}^{2}} = \frac{- 9}{- 4 {(x + 3)}^{2}}$

Step 13.6.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 9}{- 4 {(x + 3)}^{2}}$

Step 13.6.3

Simplify the right side.

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Step 13.6.3.1

Dividing two negative values results in a positive value.

$y = \frac{9}{4 {(x + 3)}^{2}}$