Algebra Examples

$y = \frac{5 (11 - y)}{3 (y + 1)}$

Step 1

Subtract $\frac{5 (11 - y)}{3 (y + 1)}$ from both sides of the equation.

$y - \frac{5 (11 - y)}{3 (y + 1)} = 0$

Step 2

Simplify $y - \frac{5 (11 - y)}{3 (y + 1)}$ .

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Step 2.1

To write $y$ as a fraction with a common denominator, multiply by $\frac{3 (y + 1)}{3 (y + 1)}$ .

$y \cdot \frac{3 (y + 1)}{3 (y + 1)} - \frac{5 (11 - y)}{3 (y + 1)} = 0$

Step 2.2

Combine $y$ and $\frac{3 (y + 1)}{3 (y + 1)}$ .

$\frac{y (3 (y + 1))}{3 (y + 1)} - \frac{5 (11 - y)}{3 (y + 1)} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{y (3 (y + 1)) - 5 (11 - y)}{3 (y + 1)} = 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Rewrite using the commutative property of multiplication.

$\frac{3 y (y + 1) - 5 (11 - y)}{3 (y + 1)} = 0$

Step 2.4.2

Apply the distributive property.

$\frac{3 y \cdot y + 3 y \cdot 1 - 5 (11 - y)}{3 (y + 1)} = 0$

Step 2.4.3

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

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Step 2.4.3.1

Move $y$ .

$\frac{3 (y \cdot y) + 3 y \cdot 1 - 5 (11 - y)}{3 (y + 1)} = 0$

Step 2.4.3.2

Multiply $y$ by $y$ .

$\frac{3 y^{2} + 3 y \cdot 1 - 5 (11 - y)}{3 (y + 1)} = 0$

Step 2.4.4

Multiply $3$ by $1$ .

$\frac{3 y^{2} + 3 y - 5 (11 - y)}{3 (y + 1)} = 0$

Step 2.4.5

Apply the distributive property.

$\frac{3 y^{2} + 3 y - 5 \cdot 11 - 5 (- y)}{3 (y + 1)} = 0$

Step 2.4.6

Multiply $- 5$ by $11$ .

$\frac{3 y^{2} + 3 y - 55 - 5 (- y)}{3 (y + 1)} = 0$

Step 2.4.7

Multiply $- 1$ by $- 5$ .

$\frac{3 y^{2} + 3 y - 55 + 5 y}{3 (y + 1)} = 0$

Step 2.4.8

Add $3 y$ and $5 y$ .

$\frac{3 y^{2} + 8 y - 55}{3 (y + 1)} = 0$

Step 3

Set the numerator equal to zero.

$3 y^{2} + 8 y - 55 = 0$

Step 4

Solve the equation for $y$ .

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Step 4.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.2

Substitute the values $a = 3$ , $b = 8$ , and $c = - 55$ into the quadratic formula and solve for $y$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (3 \cdot - 55)}}{2 \cdot 3}$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Raise $8$ to the power of $2$ .

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 55}}{2 \cdot 3}$

Step 4.3.1.2

Multiply $- 4 \cdot 3 \cdot - 55$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{- 8 \pm \sqrt{64 - 12 \cdot - 55}}{2 \cdot 3}$

Step 4.3.1.2.2

Multiply $- 12$ by $- 55$ .

$y = \frac{- 8 \pm \sqrt{64 + 660}}{2 \cdot 3}$

Step 4.3.1.3

Add $64$ and $660$ .

$y = \frac{- 8 \pm \sqrt{724}}{2 \cdot 3}$

Step 4.3.1.4

Rewrite $724$ as $2^{2} \cdot 181$ .

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Step 4.3.1.4.1

Factor $4$ out of $724$ .

$y = \frac{- 8 \pm \sqrt{4 (181)}}{2 \cdot 3}$

Step 4.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 8 \pm \sqrt{2^{2} \cdot 181}}{2 \cdot 3}$

Step 4.3.1.5

Pull terms out from under the radical.

$y = \frac{- 8 \pm 2 \sqrt{181}}{2 \cdot 3}$

Step 4.3.2

Multiply $2$ by $3$ .

$y = \frac{- 8 \pm 2 \sqrt{181}}{6}$

Step 4.3.3

Simplify $\frac{- 8 \pm 2 \sqrt{181}}{6}$ .

$y = \frac{- 4 \pm \sqrt{181}}{3}$

Step 4.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Raise $8$ to the power of $2$ .

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 55}}{2 \cdot 3}$

Step 4.4.1.2

Multiply $- 4 \cdot 3 \cdot - 55$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{- 8 \pm \sqrt{64 - 12 \cdot - 55}}{2 \cdot 3}$

Step 4.4.1.2.2

Multiply $- 12$ by $- 55$ .

$y = \frac{- 8 \pm \sqrt{64 + 660}}{2 \cdot 3}$

Step 4.4.1.3

Add $64$ and $660$ .

$y = \frac{- 8 \pm \sqrt{724}}{2 \cdot 3}$

Step 4.4.1.4

Rewrite $724$ as $2^{2} \cdot 181$ .

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Step 4.4.1.4.1

Factor $4$ out of $724$ .

$y = \frac{- 8 \pm \sqrt{4 (181)}}{2 \cdot 3}$

Step 4.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 8 \pm \sqrt{2^{2} \cdot 181}}{2 \cdot 3}$

Step 4.4.1.5

Pull terms out from under the radical.

$y = \frac{- 8 \pm 2 \sqrt{181}}{2 \cdot 3}$

Step 4.4.2

Multiply $2$ by $3$ .

$y = \frac{- 8 \pm 2 \sqrt{181}}{6}$

Step 4.4.3

Simplify $\frac{- 8 \pm 2 \sqrt{181}}{6}$ .

$y = \frac{- 4 \pm \sqrt{181}}{3}$

Step 4.4.4

Change the $\pm$ to $+$ .

$y = \frac{- 4 + \sqrt{181}}{3}$

Step 4.4.5

Rewrite $- 4$ as $- 1 (4)$ .

$y = \frac{- 1 \cdot 4 + \sqrt{181}}{3}$

Step 4.4.6

Factor $- 1$ out of $\sqrt{181}$ .

$y = \frac{- 1 \cdot 4 - 1 (- \sqrt{181})}{3}$

Step 4.4.7

Factor $- 1$ out of $- 1 (4) - 1 (- \sqrt{181})$ .

$y = \frac{- 1 (4 - \sqrt{181})}{3}$

Step 4.4.8

Move the negative in front of the fraction.

$y = - \frac{4 - \sqrt{181}}{3}$

Step 4.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $8$ to the power of $2$ .

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 55}}{2 \cdot 3}$

Step 4.5.1.2

Multiply $- 4 \cdot 3 \cdot - 55$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{- 8 \pm \sqrt{64 - 12 \cdot - 55}}{2 \cdot 3}$

Step 4.5.1.2.2

Multiply $- 12$ by $- 55$ .

$y = \frac{- 8 \pm \sqrt{64 + 660}}{2 \cdot 3}$

Step 4.5.1.3

Add $64$ and $660$ .

$y = \frac{- 8 \pm \sqrt{724}}{2 \cdot 3}$

Step 4.5.1.4

Rewrite $724$ as $2^{2} \cdot 181$ .

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Step 4.5.1.4.1

Factor $4$ out of $724$ .

$y = \frac{- 8 \pm \sqrt{4 (181)}}{2 \cdot 3}$

Step 4.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 8 \pm \sqrt{2^{2} \cdot 181}}{2 \cdot 3}$

Step 4.5.1.5

Pull terms out from under the radical.

$y = \frac{- 8 \pm 2 \sqrt{181}}{2 \cdot 3}$

Step 4.5.2

Multiply $2$ by $3$ .

$y = \frac{- 8 \pm 2 \sqrt{181}}{6}$

Step 4.5.3

Simplify $\frac{- 8 \pm 2 \sqrt{181}}{6}$ .

$y = \frac{- 4 \pm \sqrt{181}}{3}$

Step 4.5.4

Change the $\pm$ to $-$ .

$y = \frac{- 4 - \sqrt{181}}{3}$

Step 4.5.5

Rewrite $- 4$ as $- 1 (4)$ .

$y = \frac{- 1 \cdot 4 - \sqrt{181}}{3}$

Step 4.5.6

Factor $- 1$ out of $- \sqrt{181}$ .

$y = \frac{- 1 \cdot 4 - (\sqrt{181})}{3}$

Step 4.5.7

Factor $- 1$ out of $- 1 (4) - (\sqrt{181})$ .

$y = \frac{- 1 (4 + \sqrt{181})}{3}$

Step 4.5.8

Move the negative in front of the fraction.

$y = - \frac{4 + \sqrt{181}}{3}$

Step 4.6

The final answer is the combination of both solutions.

$y = - \frac{4 - \sqrt{181}}{3}, - \frac{4 + \sqrt{181}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{4 - \sqrt{181}}{3}, - \frac{4 + \sqrt{181}}{3}$

Decimal Form:

$y = 3.15120801 \dots, - 5.81787468 \dots$