Finite Math Examples

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

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

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

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

Step 2

Solve the equation for

v

$v$ .

Tap for more steps...

Step 2.1

Simplify

(4 - v) (v - 6)

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

Tap for more steps...

Step 2.1.1

Rewrite.

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

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

Step 2.1.2

Simplify by adding zeros.

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

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

Step 2.1.3

Expand

(4 - v) (v - 6)

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

Tap for more steps...

Step 2.1.3.1

Apply the distributive property.

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

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

Step 2.1.3.2

Apply the distributive property.

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

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

Step 2.1.3.3

Apply the distributive property.

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

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

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

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

Step 2.1.4

Simplify and combine like terms.

Tap for more steps...

Step 2.1.4.1

Simplify each term.

Tap for more steps...

Step 2.1.4.1.1

Multiply

4

$4$ by

- 6

$- 6$ .

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

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

Step 2.1.4.1.2

Multiply

v

$v$ by

v

$v$ by adding the exponents.

Tap for more steps...

Step 2.1.4.1.2.1

Move

v

$v$ .

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

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

Step 2.1.4.1.2.2

Multiply

v

$v$ by

v

$v$ .

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

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

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

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

Step 2.1.4.1.3

Multiply

- 6

$- 6$ by

- 1

$- 1$ .

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

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

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

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

Step 2.1.4.2

Add

4 v

$4 v$ and

6 v

$6 v$ .

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

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

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

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

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

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

Step 2.2

Simplify

(6 - v) \cdot 2

$(6 - v) \cdot 2$ .

Tap for more steps...

Step 2.2.1

Apply the distributive property.

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

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

Step 2.2.2

Multiply.

Tap for more steps...

Step 2.2.2.1

Multiply

6

$6$ by

2

$2$ .

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

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

Step 2.2.2.2

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

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

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

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

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

Step 2.3

Move all terms containing

v

$v$ to the left side of the equation.

Tap for more steps...

Step 2.3.1

Add

2 v

$2 v$ to both sides of the equation.

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

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

Step 2.3.2

Add

10 v

$10 v$ and

2 v

$2 v$ .

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

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

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

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

Step 2.4

Subtract

12

$12$ from both sides of the equation.

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

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

Step 2.5

Subtract

12

$12$ from

- 24

$- 24$ .

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

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

Step 2.6

Factor the left side of the equation.

Tap for more steps...

Step 2.6.1

Factor

- 1

$- 1$ out of

12 v - v^{2} - 36

$12 v - v^{2} - 36$ .

Tap for more steps...

Step 2.6.1.1

Reorder

12 v

$12 v$ and

- v^{2}

$- v^{2}$ .

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

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

Step 2.6.1.2

Factor

- 1

$- 1$ out of

- v^{2}

$- v^{2}$ .

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

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

Step 2.6.1.3

Factor

- 1

$- 1$ out of

12 v

$12 v$ .

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

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

Step 2.6.1.4

Rewrite

- 36

$- 36$ as

- 1 (36)

$- 1 (36)$ .

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

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

Step 2.6.1.5

Factor

- 1

$- 1$ out of

- (v^{2}) - (- 12 v)

$- (v^{2}) - (- 12 v)$ .

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

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

Step 2.6.1.6

Factor

- 1

$- 1$ out of

- (v^{2} - 12 v) - 1 (36)

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

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

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

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

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

Step 2.6.2

Factor using the perfect square rule.

Tap for more steps...

Step 2.6.2.1

Rewrite

36

$36$ as

6^{2}

$6^{2}$ .

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

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

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

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

Step 2.6.2.3

Rewrite the polynomial.

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

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

Step 2.6.2.4

Factor using the perfect square trinomial rule

a^{2} - 2 a b + b^{2} = {(a - b)}^{2}

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

a = v

$a = v$ and

b = 6

$b = 6$ .

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

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

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

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

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

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

Step 2.7

Divide each term in

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

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

- 1

$- 1$ and simplify.

Tap for more steps...

Step 2.7.1

Divide each term in

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

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

- 1

$- 1$ .

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

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

Step 2.7.2

Simplify the left side.

Tap for more steps...

Step 2.7.2.1

Dividing two negative values results in a positive value.

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

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

Step 2.7.2.2

Divide

{(v - 6)}^{2}

${(v - 6)}^{2}$ by

1

$1$ .

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

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

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

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

Step 2.7.3

Simplify the right side.

Tap for more steps...

Step 2.7.3.1

Divide

0

$0$ by

- 1

$- 1$ .

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

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

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

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

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

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

Step 2.8

Set the

v - 6

$v - 6$ equal to

0

$0$ .

v - 6 = 0

$v - 6 = 0$

Step 2.9

Add

6

$6$ to both sides of the equation.

v = 6

$v = 6$

v = 6

$v = 6$

Step 3

Exclude the solutions that do not make

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

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

No solution

$No solution$