Algebra Examples

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

Step 1

Factor each term.

Tap for more steps...

Step 1.1

Simplify the numerator.

Tap for more steps...

Step 1.1.1

To write $y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.1.2

Combine $y$ and $\frac{2}{2}$ .

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

Step 1.1.3

Combine the numerators over the common denominator.

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

Step 1.1.4

Move $2$ to the left of $y$ .

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

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.3

Multiply $\frac{2 y - 1}{2} \cdot \frac{1}{3}$ .

Tap for more steps...

Step 1.3.1

Multiply $\frac{2 y - 1}{2}$ by $\frac{1}{3}$ .

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

Step 1.3.2

Multiply $2$ by $3$ .

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

Step 1.4

Factor $2$ out of $2 y + 2$ .

Tap for more steps...

Step 1.4.1

Factor $2$ out of $2 y$ .

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

Step 1.4.2

Factor $2$ out of $2$ .

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

Step 1.4.3

Factor $2$ out of $2 (y) + 2 (1)$ .

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

Step 1.5

Simplify the numerator.

Tap for more steps...

Step 1.5.1

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.5.2

Combine $2 y$ and $\frac{2}{2}$ .

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

Step 1.5.3

Combine the numerators over the common denominator.

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

Step 1.5.4

Multiply $2$ by $2$ .

$\frac{2 y - 1}{6} - \frac{2 (y + 1)}{2 y + 1} = \frac{\frac{4 y + 1}{2}}{6}$

Step 1.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 y - 1}{6} - \frac{2 (y + 1)}{2 y + 1} = \frac{4 y + 1}{2} \cdot \frac{1}{6}$

Step 1.7

Multiply $\frac{4 y + 1}{2} \cdot \frac{1}{6}$ .

Tap for more steps...

Step 1.7.1

Multiply $\frac{4 y + 1}{2}$ by $\frac{1}{6}$ .

$\frac{2 y - 1}{6} - \frac{2 (y + 1)}{2 y + 1} = \frac{4 y + 1}{2 \cdot 6}$

Step 1.7.2

Multiply $2$ by $6$ .

$\frac{2 y - 1}{6} - \frac{2 (y + 1)}{2 y + 1} = \frac{4 y + 1}{12}$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

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

$6, 2 y + 1, 12$

Step 2.2

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 2.3

$6$ has factors of $2$ and $3$ .

$2 \cdot 3$

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 prime factors for $12$ are $2 \cdot 2 \cdot 3$ .

Tap for more steps...

Step 2.5.1

$12$ has factors of $2$ and $6$ .

$2 \cdot 6$

Step 2.5.2

$6$ has factors of $2$ and $3$ .

$2 \cdot 2 \cdot 3$

Step 2.6

Multiply $2 \cdot 2 \cdot 3$ .

Tap for more steps...

Step 2.6.1

Multiply $2$ by $2$ .

$4 \cdot 3$

Step 2.6.2

Multiply $4$ by $3$ .

$12$

Step 2.7

The factor for $2 y + 1$ is $2 y + 1$ itself.

$(2 y + 1) = 2 y + 1$

$(2 y + 1)$ occurs $1$ time.

Step 2.8

The LCM of $2 y + 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$2 y + 1$

Step 2.9

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

$12 (2 y + 1)$

Step 3

Multiply each term in $\frac{2 y - 1}{6} - \frac{2 (y + 1)}{2 y + 1} = \frac{4 y + 1}{12}$ by $12 (2 y + 1)$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{2 y - 1}{6} - \frac{2 (y + 1)}{2 y + 1} = \frac{4 y + 1}{12}$ by $12 (2 y + 1)$ .

$\frac{2 y - 1}{6} (12 (2 y + 1)) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Rewrite using the commutative property of multiplication.

$12 \frac{2 y - 1}{6} (2 y + 1) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.2

Cancel the common factor of $6$ .

Tap for more steps...

Step 3.2.1.2.1

Factor $6$ out of $12$ .

$6 (2) \frac{2 y - 1}{6} (2 y + 1) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.2.2

Cancel the common factor.

$6 \cdot 2 \frac{2 y - 1}{6} (2 y + 1) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.2.3

Rewrite the expression.

$2 (2 y - 1) (2 y + 1) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.3

Apply the distributive property.

$(2 (2 y) + 2 \cdot - 1) (2 y + 1) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.4

Multiply $2$ by $2$ .

$(4 y + 2 \cdot - 1) (2 y + 1) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.5

Multiply $2$ by $- 1$ .

$(4 y - 2) (2 y + 1) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.6

Expand $(4 y - 2) (2 y + 1)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.6.1

Apply the distributive property.

$4 y (2 y + 1) - 2 (2 y + 1) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.6.2

Apply the distributive property.

$4 y (2 y) + 4 y \cdot 1 - 2 (2 y + 1) - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.6.3

Apply the distributive property.

$4 y (2 y) + 4 y \cdot 1 - 2 (2 y) - 2 \cdot 1 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.7

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.7.1

Simplify each term.

Tap for more steps...

Step 3.2.1.7.1.1

Rewrite using the commutative property of multiplication.

$4 \cdot 2 y \cdot y + 4 y \cdot 1 - 2 (2 y) - 2 \cdot 1 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.7.1.2

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

Tap for more steps...

Step 3.2.1.7.1.2.1

Move $y$ .

$4 \cdot 2 (y \cdot y) + 4 y \cdot 1 - 2 (2 y) - 2 \cdot 1 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.7.1.2.2

Multiply $y$ by $y$ .

$4 \cdot 2 y^{2} + 4 y \cdot 1 - 2 (2 y) - 2 \cdot 1 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.7.1.3

Multiply $4$ by $2$ .

$8 y^{2} + 4 y \cdot 1 - 2 (2 y) - 2 \cdot 1 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.7.1.4

Multiply $4$ by $1$ .

$8 y^{2} + 4 y - 2 (2 y) - 2 \cdot 1 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.7.1.5

Multiply $2$ by $- 2$ .

$8 y^{2} + 4 y - 4 y - 2 \cdot 1 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.7.1.6

Multiply $- 2$ by $1$ .

$8 y^{2} + 4 y - 4 y - 2 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.7.2

Subtract $4 y$ from $4 y$ .

$8 y^{2} + 0 - 2 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.7.3

Add $8 y^{2}$ and $0$ .

$8 y^{2} - 2 - \frac{2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.8

Cancel the common factor of $2 y + 1$ .

Tap for more steps...

Step 3.2.1.8.1

Move the leading negative in $- \frac{2 (y + 1)}{2 y + 1}$ into the numerator.

$8 y^{2} - 2 + \frac{- 2 (y + 1)}{2 y + 1} (12 (2 y + 1)) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.8.2

Factor $2 y + 1$ out of $12 (2 y + 1)$ .

$8 y^{2} - 2 + \frac{- 2 (y + 1)}{2 y + 1} ((2 y + 1) \cdot 12) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.8.3

Cancel the common factor.

$8 y^{2} - 2 + \frac{- 2 (y + 1)}{2 y + 1} ((2 y + 1) \cdot 12) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.8.4

Rewrite the expression.

$8 y^{2} - 2 - 2 (y + 1) \cdot 12 = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.9

Multiply $12$ by $- 2$ .

$8 y^{2} - 2 - 24 (y + 1) = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.10

Apply the distributive property.

$8 y^{2} - 2 - 24 y - 24 \cdot 1 = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.1.11

Multiply $- 24$ by $1$ .

$8 y^{2} - 2 - 24 y - 24 = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.2.2

Subtract $24$ from $- 2$ .

$8 y^{2} - 24 y - 26 = \frac{4 y + 1}{12} (12 (2 y + 1))$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$8 y^{2} - 24 y - 26 = 12 \frac{4 y + 1}{12} (2 y + 1)$

Step 3.3.1.2

Cancel the common factor of $12$ .

Tap for more steps...

Step 3.3.1.2.1

Cancel the common factor.

$8 y^{2} - 24 y - 26 = 12 \frac{4 y + 1}{12} (2 y + 1)$

Step 3.3.1.2.2

Rewrite the expression.

$8 y^{2} - 24 y - 26 = (4 y + 1) (2 y + 1)$

Step 3.3.2

Expand $(4 y + 1) (2 y + 1)$ using the FOIL Method.

Tap for more steps...

Step 3.3.2.1

Apply the distributive property.

$8 y^{2} - 24 y - 26 = 4 y (2 y + 1) + 1 (2 y + 1)$

Step 3.3.2.2

Apply the distributive property.

$8 y^{2} - 24 y - 26 = 4 y (2 y) + 4 y \cdot 1 + 1 (2 y + 1)$

Step 3.3.2.3

Apply the distributive property.

$8 y^{2} - 24 y - 26 = 4 y (2 y) + 4 y \cdot 1 + 1 (2 y) + 1 \cdot 1$

Step 3.3.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.3.1.1

Rewrite using the commutative property of multiplication.

$8 y^{2} - 24 y - 26 = 4 \cdot 2 y \cdot y + 4 y \cdot 1 + 1 (2 y) + 1 \cdot 1$

Step 3.3.3.1.2

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

Tap for more steps...

Step 3.3.3.1.2.1

Move $y$ .

$8 y^{2} - 24 y - 26 = 4 \cdot 2 (y \cdot y) + 4 y \cdot 1 + 1 (2 y) + 1 \cdot 1$

Step 3.3.3.1.2.2

Multiply $y$ by $y$ .

$8 y^{2} - 24 y - 26 = 4 \cdot 2 y^{2} + 4 y \cdot 1 + 1 (2 y) + 1 \cdot 1$

Step 3.3.3.1.3

Multiply $4$ by $2$ .

$8 y^{2} - 24 y - 26 = 8 y^{2} + 4 y \cdot 1 + 1 (2 y) + 1 \cdot 1$

Step 3.3.3.1.4

Multiply $4$ by $1$ .

$8 y^{2} - 24 y - 26 = 8 y^{2} + 4 y + 1 (2 y) + 1 \cdot 1$

Step 3.3.3.1.5

Multiply $2 y$ by $1$ .

$8 y^{2} - 24 y - 26 = 8 y^{2} + 4 y + 2 y + 1 \cdot 1$

Step 3.3.3.1.6

Multiply $1$ by $1$ .

$8 y^{2} - 24 y - 26 = 8 y^{2} + 4 y + 2 y + 1$

Step 3.3.3.2

Add $4 y$ and $2 y$ .

$8 y^{2} - 24 y - 26 = 8 y^{2} + 6 y + 1$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Subtract $8 y^{2}$ from both sides of the equation.

$8 y^{2} - 24 y - 26 - 8 y^{2} = 6 y + 1$

Step 4.1.2

Subtract $6 y$ from both sides of the equation.

$8 y^{2} - 24 y - 26 - 8 y^{2} - 6 y = 1$

Step 4.1.3

Combine the opposite terms in $8 y^{2} - 24 y - 26 - 8 y^{2} - 6 y$ .

Tap for more steps...

Step 4.1.3.1

Subtract $8 y^{2}$ from $8 y^{2}$ .

$- 24 y - 26 + 0 - 6 y = 1$

Step 4.1.3.2

Add $- 24 y - 26$ and $0$ .

$- 24 y - 26 - 6 y = 1$

Step 4.1.4

Subtract $6 y$ from $- 24 y$ .

$- 30 y - 26 = 1$

Step 4.2

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 4.2.1

Add $26$ to both sides of the equation.

$- 30 y = 1 + 26$

Step 4.2.2

Add $1$ and $26$ .

$- 30 y = 27$

Step 4.3

Divide each term in $- 30 y = 27$ by $- 30$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $- 30 y = 27$ by $- 30$ .

$\frac{- 30 y}{- 30} = \frac{27}{- 30}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $- 30$ .

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 30 y}{- 30} = \frac{27}{- 30}$

Step 4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{27}{- 30}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Cancel the common factor of $27$ and $- 30$ .

Tap for more steps...

Step 4.3.3.1.1

Factor $3$ out of $27$ .

$y = \frac{3 (9)}{- 30}$

Step 4.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 4.3.3.1.2.1

Factor $3$ out of $- 30$ .

$y = \frac{3 \cdot 9}{3 \cdot - 10}$

Step 4.3.3.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 9}{3 \cdot - 10}$

Step 4.3.3.1.2.3

Rewrite the expression.

$y = \frac{9}{- 10}$

Step 4.3.3.2

Move the negative in front of the fraction.

$y = - \frac{9}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{9}{10}$

Decimal Form:

$y = - 0.9$