Finite Math Examples

$\frac{4 - v}{6 - v} = \frac{2}{v - 6}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(4 - v) (v - 6) = (6 - v) \cdot 2$

Step 2

Solve the equation for $v$ .

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Step 2.1

Simplify $(4 - v) (v - 6)$ .

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Step 2.1.1

Rewrite.

$0 + 0 + (4 - v) (v - 6) = (6 - v) \cdot 2$

Step 2.1.2

Simplify by adding zeros.

$(4 - v) (v - 6) = (6 - v) \cdot 2$

Step 2.1.3

Expand $(4 - v) (v - 6)$ using the FOIL Method.

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Step 2.1.3.1

Apply the distributive property.

$4 (v - 6) - v (v - 6) = (6 - v) \cdot 2$

Step 2.1.3.2

Apply the distributive property.

$4 v + 4 \cdot - 6 - v (v - 6) = (6 - v) \cdot 2$

Step 2.1.3.3

Apply the distributive property.

$4 v + 4 \cdot - 6 - v \cdot v - v \cdot - 6 = (6 - v) \cdot 2$

Step 2.1.4

Simplify and combine like terms.

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Step 2.1.4.1

Simplify each term.

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Step 2.1.4.1.1

Multiply $4$ by $- 6$ .

$4 v - 24 - v \cdot v - v \cdot - 6 = (6 - v) \cdot 2$

Step 2.1.4.1.2

Multiply $v$ by $v$ by adding the exponents.

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Step 2.1.4.1.2.1

Move $v$ .

$4 v - 24 - (v \cdot v) - v \cdot - 6 = (6 - v) \cdot 2$

Step 2.1.4.1.2.2

Multiply $v$ by $v$ .

$4 v - 24 - v^{2} - v \cdot - 6 = (6 - v) \cdot 2$

Step 2.1.4.1.3

Multiply $- 6$ by $- 1$ .

$4 v - 24 - v^{2} + 6 v = (6 - v) \cdot 2$

Step 2.1.4.2

Add $4 v$ and $6 v$ .

$10 v - 24 - v^{2} = (6 - v) \cdot 2$

Step 2.2

Simplify $(6 - v) \cdot 2$ .

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Step 2.2.1

Apply the distributive property.

$10 v - 24 - v^{2} = 6 \cdot 2 - v \cdot 2$

Step 2.2.2

Multiply.

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Step 2.2.2.1

Multiply $6$ by $2$ .

$10 v - 24 - v^{2} = 12 - v \cdot 2$

Step 2.2.2.2

Multiply $2$ by $- 1$ .

$10 v - 24 - v^{2} = 12 - 2 v$

Step 2.3

Move all terms containing $v$ to the left side of the equation.

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Step 2.3.1

Add $2 v$ to both sides of the equation.

$10 v - 24 - v^{2} + 2 v = 12$

Step 2.3.2

Add $10 v$ and $2 v$ .

$12 v - 24 - v^{2} = 12$

Step 2.4

Subtract $12$ from both sides of the equation.

$12 v - 24 - v^{2} - 12 = 0$

Step 2.5

Subtract $12$ from $- 24$ .

$12 v - v^{2} - 36 = 0$

Step 2.6

Factor the left side of the equation.

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Step 2.6.1

Factor $- 1$ out of $12 v - v^{2} - 36$ .

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Step 2.6.1.1

Reorder $12 v$ and $- v^{2}$ .

$- v^{2} + 12 v - 36 = 0$

Step 2.6.1.2

Factor $- 1$ out of $- v^{2}$ .

$- (v^{2}) + 12 v - 36 = 0$

Step 2.6.1.3

Factor $- 1$ out of $12 v$ .

$- (v^{2}) - (- 12 v) - 36 = 0$

Step 2.6.1.4

Rewrite $- 36$ as $- 1 (36)$ .

$- (v^{2}) - (- 12 v) - 1 \cdot 36 = 0$

Step 2.6.1.5

Factor $- 1$ out of $- (v^{2}) - (- 12 v)$ .

$- (v^{2} - 12 v) - 1 \cdot 36 = 0$

Step 2.6.1.6

Factor $- 1$ out of $- (v^{2} - 12 v) - 1 (36)$ .

$- (v^{2} - 12 v + 36) = 0$

Step 2.6.2

Factor using the perfect square rule.

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Step 2.6.2.1

Rewrite $36$ as $6^{2}$ .

$- (v^{2} - 12 v + 6^{2}) = 0$

Step 2.6.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 v = 2 \cdot v \cdot 6$

Step 2.6.2.3

Rewrite the polynomial.

$- (v^{2} - 2 \cdot v \cdot 6 + 6^{2}) = 0$

Step 2.6.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = v$ and $b = 6$ .

$- {(v - 6)}^{2} = 0$

Step 2.7

Divide each term in $- {(v - 6)}^{2} = 0$ by $- 1$ and simplify.

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Step 2.7.1

Divide each term in $- {(v - 6)}^{2} = 0$ by $- 1$ .

$\frac{- {(v - 6)}^{2}}{- 1} = \frac{0}{- 1}$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Dividing two negative values results in a positive value.

$\frac{{(v - 6)}^{2}}{1} = \frac{0}{- 1}$

Step 2.7.2.2

Divide ${(v - 6)}^{2}$ by $1$ .

${(v - 6)}^{2} = \frac{0}{- 1}$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

Divide $0$ by $- 1$ .

${(v - 6)}^{2} = 0$

Step 2.8

Set the $v - 6$ equal to $0$ .

$v - 6 = 0$

Step 2.9

Add $6$ to both sides of the equation.

$v = 6$

Step 3

Exclude the solutions that do not make $\frac{4 - v}{6 - v} = \frac{2}{v - 6}$ true.

$No solution$