Basic Math Examples

$\frac{12}{y} = \frac{6}{2 y - 6} + 1$

Step 1

Factor each term.

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Step 1.1

Factor $2$ out of $2 y - 6$ .

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Step 1.1.1

Factor $2$ out of $2 y$ .

$\frac{12}{y} = \frac{6}{2 (y) - 6} + 1$

Step 1.1.2

Factor $2$ out of $- 6$ .

$\frac{12}{y} = \frac{6}{2 y + 2 \cdot - 3} + 1$

Step 1.1.3

Factor $2$ out of $2 y + 2 \cdot - 3$ .

$\frac{12}{y} = \frac{6}{2 (y - 3)} + 1$

Step 1.2

Reduce the expression $\frac{6}{2 (y - 3)}$ by cancelling the common factors.

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Step 1.2.1

Factor $2$ out of $6$ .

$\frac{12}{y} = \frac{2 \cdot 3}{2 (y - 3)} + 1$

Step 1.2.2

Cancel the common factor.

$\frac{12}{y} = \frac{2 \cdot 3}{2 (y - 3)} + 1$

Step 1.2.3

Rewrite the expression.

$\frac{12}{y} = \frac{3}{y - 3} + 1$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, y - 3, 1$

Step 2.2

Since $y, y - 3, 1$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $y, y - 3, 1$ are:

1. Find the LCM for the numeric part $1, 1, 1$ .

2. Find the LCM for the variable part $y^{1}$ .

3. Find the LCM for the compound variable part $y - 3$ .

4. Multiply each LCM together.

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.6

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 2.7

The LCM of $y^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y$

Step 2.8

The factor for $y - 3$ is $y - 3$ itself.

$(y - 3) = y - 3$

$(y - 3)$ occurs $1$ time.

Step 2.9

The LCM of $y - 3$ is the result of multiplying all factors the greatest number of times they occur in either term.

$y - 3$

Step 2.10

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$y (y - 3)$

Step 3

Multiply each term in $\frac{12}{y} = \frac{3}{y - 3} + 1$ by $y (y - 3)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{12}{y} = \frac{3}{y - 3} + 1$ by $y (y - 3)$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $y$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Multiply $12$ by $- 3$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $y - 3$ .

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Step 3.3.1.1.1

Factor $y - 3$ out of $y (y - 3)$ .

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

Step 3.3.1.1.2

Cancel the common factor.

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

Step 3.3.1.1.3

Rewrite the expression.

$12 y - 36 = 3 y + 1 (y (y - 3))$

Step 3.3.1.2

Multiply $y (y - 3)$ by $1$ .

$12 y - 36 = 3 y + y (y - 3)$

Step 3.3.1.3

Apply the distributive property.

$12 y - 36 = 3 y + y \cdot y + y \cdot - 3$

Step 3.3.1.4

Multiply $y$ by $y$ .

$12 y - 36 = 3 y + y^{2} + y \cdot - 3$

Step 3.3.1.5

Move $- 3$ to the left of $y$ .

$12 y - 36 = 3 y + y^{2} - 3 y$

Step 3.3.2

Combine the opposite terms in $3 y + y^{2} - 3 y$ .

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Step 3.3.2.1

Subtract $3 y$ from $3 y$ .

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

Step 3.3.2.2

Add $y^{2}$ and $0$ .

$12 y - 36 = y^{2}$

Step 4

Solve the equation.

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Step 4.1

Subtract $y^{2}$ from both sides of the equation.

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

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

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

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $- 36$ .

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

Step 4.2.1.1.2

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

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

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

Step 4.2.1.5

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

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

Step 4.2.1.6

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

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

Step 4.2.2

Factor using the perfect square rule.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

$12 y = 2 \cdot y \cdot 6$

Step 4.2.2.3

Rewrite the polynomial.

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

Step 4.2.2.4

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

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

Step 4.3.2.2

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

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

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $0$ by $- 1$ .

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

Step 4.4

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

$y - 6 = 0$

Step 4.5

Add $6$ to both sides of the equation.

$y = 6$