Basic Math Examples

$| y - 1 | = \frac{2}{y}$

Step 1

Multiply both sides by $y$ .

$| y - 1 | y = \frac{2}{y} y$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Reorder factors in $| y - 1 | y$ .

$y | y - 1 | = \frac{2}{y} y$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $y$ .

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Step 2.2.1.1

Cancel the common factor.

$y | y - 1 | = \frac{2}{y} y$

Step 2.2.1.2

Rewrite the expression.

$y | y - 1 | = 2$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $y | y - 1 | = 2$ by $y$ and simplify.

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Step 3.1.1

Divide each term in $y | y - 1 | = 2$ by $y$ .

$\frac{y | y - 1 |}{y} = \frac{2}{y}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $y$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{y | y - 1 |}{y} = \frac{2}{y}$

Step 3.1.2.1.2

Divide $| y - 1 |$ by $1$ .

$| y - 1 | = \frac{2}{y}$

Step 3.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y - 1 = \pm \frac{2}{y}$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 3.3.2

Add $1$ to both sides of the equation.

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

Step 3.3.3

Find the LCD of the terms in the equation.

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Step 3.3.3.1

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

$1, y, 1$

Step 3.3.3.2

The LCM of one and any expression is the expression.

$y$

Step 3.3.4

Multiply each term in $y = \frac{2}{y} + 1$ by $y$ to eliminate the fractions.

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Step 3.3.4.1

Multiply each term in $y = \frac{2}{y} + 1$ by $y$ .

$y \cdot y = \frac{2}{y} y + 1 y$

Step 3.3.4.2

Simplify the left side.

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Step 3.3.4.2.1

Multiply $y$ by $y$ .

$y^{2} = \frac{2}{y} y + 1 y$

Step 3.3.4.3

Simplify the right side.

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Step 3.3.4.3.1

Simplify each term.

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Step 3.3.4.3.1.1

Cancel the common factor of $y$ .

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Step 3.3.4.3.1.1.1

Cancel the common factor.

$y^{2} = \frac{2}{y} y + 1 y$

Step 3.3.4.3.1.1.2

Rewrite the expression.

$y^{2} = 2 + 1 y$

Step 3.3.4.3.1.2

Multiply $y$ by $1$ .

$y^{2} = 2 + y$

Step 3.3.5

Solve the equation.

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Step 3.3.5.1

Subtract $y$ from both sides of the equation.

$y^{2} - y = 2$

Step 3.3.5.2

Subtract $2$ from both sides of the equation.

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

Step 3.3.5.3

Factor $y^{2} - y - 2$ using the AC method.

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Step 3.3.5.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 3.3.5.3.2

Write the factored form using these integers.

$(y - 2) (y + 1) = 0$

Step 3.3.5.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 2 = 0$

$y + 1 = 0$

Step 3.3.5.5

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 3.3.5.5.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Step 3.3.5.5.2

Add $2$ to both sides of the equation.

$y = 2$

Step 3.3.5.6

Set $y + 1$ equal to $0$ and solve for $y$ .

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Step 3.3.5.6.1

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Step 3.3.5.6.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 3.3.5.7

The final solution is all the values that make $(y - 2) (y + 1) = 0$ true.

$y = 2, - 1$

Step 3.3.6

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.3.7

Add $1$ to both sides of the equation.

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

Step 3.3.8

Find the LCD of the terms in the equation.

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Step 3.3.8.1

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

$1, y, 1$

Step 3.3.8.2

The LCM of one and any expression is the expression.

$y$

Step 3.3.9

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

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Step 3.3.9.1

Multiply each term in $y = - \frac{2}{y} + 1$ by $y$ .

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

Step 3.3.9.2

Simplify the left side.

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Step 3.3.9.2.1

Multiply $y$ by $y$ .

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

Step 3.3.9.3

Simplify the right side.

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Step 3.3.9.3.1

Simplify each term.

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Step 3.3.9.3.1.1

Cancel the common factor of $y$ .

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Step 3.3.9.3.1.1.1

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

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

Step 3.3.9.3.1.1.2

Cancel the common factor.

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

Step 3.3.9.3.1.1.3

Rewrite the expression.

$y^{2} = - 2 + 1 y$

Step 3.3.9.3.1.2

Multiply $y$ by $1$ .

$y^{2} = - 2 + y$

Step 3.3.10

Solve the equation.

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Step 3.3.10.1

Subtract $y$ from both sides of the equation.

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

Step 3.3.10.2

Add $2$ to both sides of the equation.

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

Step 3.3.10.3

Use the quadratic formula to find the solutions.

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

Step 3.3.10.4

Substitute the values $a = 1$ , $b = - 1$ , and $c = 2$ into the quadratic formula and solve for $y$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 3.3.10.5

Simplify.

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Step 3.3.10.5.1

Simplify the numerator.

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Step 3.3.10.5.1.1

Raise $- 1$ to the power of $2$ .

$y = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 3.3.10.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 3.3.10.5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{1 \pm \sqrt{1 - 4 \cdot 2}}{2 \cdot 1}$

Step 3.3.10.5.1.2.2

Multiply $- 4$ by $2$ .

$y = \frac{1 \pm \sqrt{1 - 8}}{2 \cdot 1}$

Step 3.3.10.5.1.3

Subtract $8$ from $1$ .

$y = \frac{1 \pm \sqrt{- 7}}{2 \cdot 1}$

Step 3.3.10.5.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Step 3.3.10.5.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$y = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

Step 3.3.10.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{1 \pm i \sqrt{7}}{2 \cdot 1}$

Step 3.3.10.5.2

Multiply $2$ by $1$ .

$y = \frac{1 \pm i \sqrt{7}}{2}$

Step 3.3.10.6

The final answer is the combination of both solutions.

$y = \frac{1 + i \sqrt{7}}{2}, \frac{1 - i \sqrt{7}}{2}$

Step 3.3.11

The complete solution is the result of both the positive and negative portions of the solution.

$y = 2, - 1, \frac{1 + i \sqrt{7}}{2}, \frac{1 - i \sqrt{7}}{2}$