Algebra Examples

$\frac{6}{x} - \frac{5}{y} = 2 x y$

Step 1

Solve the equation for $y$ .

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Step 1.1

Subtract $\frac{6}{x}$ from both sides of the equation.

$- \frac{5}{y} = 2 x y - \frac{6}{x}$

Step 1.2

Find the LCD of the terms in the equation.

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Step 1.2.1

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

$y, 1, x$

Step 1.2.2

Since $y, 1, x$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $y^{1}, x^{1}$ .

Step 1.2.3

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 1.2.4

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

Not prime

Step 1.2.5

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 1.2.6

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

$y^{1} = y$

$y$ occurs $1$ time.

Step 1.2.7

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

$x^{1} = x$

$x$ occurs $1$ time.

Step 1.2.8

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

$y \cdot x$

Step 1.2.9

Multiply $y$ by $x$ .

$y x$

Step 1.3

Multiply each term in $- \frac{5}{y} = 2 x y - \frac{6}{x}$ by $y x$ to eliminate the fractions.

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Step 1.3.1

Multiply each term in $- \frac{5}{y} = 2 x y - \frac{6}{x}$ by $y x$ .

$- \frac{5}{y} (y x) = 2 x y (y x) - \frac{6}{x} (y x)$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $y$ .

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Step 1.3.2.1.1

Move the leading negative in $- \frac{5}{y}$ into the numerator.

$\frac{- 5}{y} (y x) = 2 x y (y x) - \frac{6}{x} (y x)$

Step 1.3.2.1.2

Factor $y$ out of $y x$ .

$\frac{- 5}{y} (y (x)) = 2 x y (y x) - \frac{6}{x} (y x)$

Step 1.3.2.1.3

Cancel the common factor.

$\frac{- 5}{y} (y x) = 2 x y (y x) - \frac{6}{x} (y x)$

Step 1.3.2.1.4

Rewrite the expression.

$- 5 x = 2 x y (y x) - \frac{6}{x} (y x)$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.3.1.1.1

Move $x$ .

$- 5 x = 2 (x \cdot x) y \cdot y - \frac{6}{x} (y x)$

Step 1.3.3.1.1.2

Multiply $x$ by $x$ .

$- 5 x = 2 x^{2} y \cdot y - \frac{6}{x} (y x)$

Step 1.3.3.1.2

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

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Step 1.3.3.1.2.1

Move $y$ .

$- 5 x = 2 x^{2} (y \cdot y) - \frac{6}{x} (y x)$

Step 1.3.3.1.2.2

Multiply $y$ by $y$ .

$- 5 x = 2 x^{2} y^{2} - \frac{6}{x} (y x)$

Step 1.3.3.1.3

Cancel the common factor of $x$ .

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Step 1.3.3.1.3.1

Move the leading negative in $- \frac{6}{x}$ into the numerator.

$- 5 x = 2 x^{2} y^{2} + \frac{- 6}{x} (y x)$

Step 1.3.3.1.3.2

Factor $x$ out of $y x$ .

$- 5 x = 2 x^{2} y^{2} + \frac{- 6}{x} (x y)$

Step 1.3.3.1.3.3

Cancel the common factor.

$- 5 x = 2 x^{2} y^{2} + \frac{- 6}{x} (x y)$

Step 1.3.3.1.3.4

Rewrite the expression.

$- 5 x = 2 x^{2} y^{2} - 6 y$

Step 1.4

Solve the equation.

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Step 1.4.1

Rewrite the equation as $2 x^{2} y^{2} - 6 y = - 5 x$ .

$2 x^{2} y^{2} - 6 y = - 5 x$

Step 1.4.2

Add $5 x$ to both sides of the equation.

$2 x^{2} y^{2} - 6 y + 5 x = 0$

Step 1.4.3

Use the quadratic formula to find the solutions.

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

Step 1.4.4

Substitute the values $a = 2 x^{2}$ , $b = - 6$ , and $c = 5 x$ into the quadratic formula and solve for $y$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (2 x^{2} \cdot (5 x))}}{2 (2 x^{2})}$

Step 1.4.5

Simplify.

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Step 1.4.5.1

Simplify the numerator.

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Step 1.4.5.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot (2 x^{2}) \cdot (5 x)}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.2

Rewrite using the commutative property of multiplication.

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 x^{2} x)}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.4.5.1.3.1

Move $x$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 (x \cdot x^{2}))}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.4.5.1.3.2.1

Raise $x$ to the power of $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 (x \cdot x^{2}))}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 x^{1 + 2})}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.3.3

Add $1$ and $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.4

Multiply $- 4 \cdot 2 \cdot 5$ .

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Step 1.4.5.1.4.1

Multiply $- 4$ by $2$ .

$y = \frac{6 \pm \sqrt{36 - 8 \cdot (5 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.4.2

Multiply $- 8$ by $5$ .

$y = \frac{6 \pm \sqrt{36 - 40 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.5

Factor $4$ out of $36 - 40 x^{3}$ .

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Step 1.4.5.1.5.1

Factor $4$ out of $36$ .

$y = \frac{6 \pm \sqrt{4 (9) - 40 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.5.2

Factor $4$ out of $- 40 x^{3}$ .

$y = \frac{6 \pm \sqrt{4 (9) + 4 (- 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.5.3

Factor $4$ out of $4 (9) + 4 (- 10 x^{3})$ .

$y = \frac{6 \pm \sqrt{4 (9 - 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.6

Rewrite $4 (9 - 10 x^{3})$ as $2^{2} (3^{2} - 10 x^{3})$ .

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Step 1.4.5.1.6.1

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

$y = \frac{6 \pm \sqrt{2^{2} (9 - 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.6.2

Rewrite $9$ as $3^{2}$ .

$y = \frac{6 \pm \sqrt{2^{2} (3^{2} - 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.7

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{3^{2} - 10 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.5.1.8

Raise $3$ to the power of $2$ .

$y = \frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.5.2

Multiply $2$ by $2$ .

$y = \frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{2^{2} x^{2}}$

Step 1.4.5.3

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

$y = \frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{4 x^{2}}$

Step 1.4.5.4

Simplify $\frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{4 x^{2}}$ .

$y = \frac{3 \pm \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

Step 1.4.6

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

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Step 1.4.6.1

Simplify the numerator.

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Step 1.4.6.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot (2 x^{2}) \cdot (5 x)}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.2

Rewrite using the commutative property of multiplication.

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 x^{2} x)}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.4.6.1.3.1

Move $x$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 (x \cdot x^{2}))}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.4.6.1.3.2.1

Raise $x$ to the power of $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 (x \cdot x^{2}))}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 x^{1 + 2})}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.3.3

Add $1$ and $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.4

Multiply $- 4 \cdot 2 \cdot 5$ .

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Step 1.4.6.1.4.1

Multiply $- 4$ by $2$ .

$y = \frac{6 \pm \sqrt{36 - 8 \cdot (5 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.4.2

Multiply $- 8$ by $5$ .

$y = \frac{6 \pm \sqrt{36 - 40 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.5

Factor $4$ out of $36 - 40 x^{3}$ .

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Step 1.4.6.1.5.1

Factor $4$ out of $36$ .

$y = \frac{6 \pm \sqrt{4 (9) - 40 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.5.2

Factor $4$ out of $- 40 x^{3}$ .

$y = \frac{6 \pm \sqrt{4 (9) + 4 (- 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.5.3

Factor $4$ out of $4 (9) + 4 (- 10 x^{3})$ .

$y = \frac{6 \pm \sqrt{4 (9 - 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.6

Rewrite $4 (9 - 10 x^{3})$ as $2^{2} (3^{2} - 10 x^{3})$ .

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Step 1.4.6.1.6.1

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

$y = \frac{6 \pm \sqrt{2^{2} (9 - 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.6.2

Rewrite $9$ as $3^{2}$ .

$y = \frac{6 \pm \sqrt{2^{2} (3^{2} - 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.7

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{3^{2} - 10 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.6.1.8

Raise $3$ to the power of $2$ .

$y = \frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.6.2

Multiply $2$ by $2$ .

$y = \frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{2^{2} x^{2}}$

Step 1.4.6.3

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

$y = \frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{4 x^{2}}$

Step 1.4.6.4

Simplify $\frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{4 x^{2}}$ .

$y = \frac{3 \pm \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

Step 1.4.6.5

Change the $\pm$ to $+$ .

$y = \frac{3 + \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

Step 1.4.7

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

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Step 1.4.7.1

Simplify the numerator.

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Step 1.4.7.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot (2 x^{2}) \cdot (5 x)}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.2

Rewrite using the commutative property of multiplication.

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 x^{2} x)}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.4.7.1.3.1

Move $x$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 (x \cdot x^{2}))}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.4.7.1.3.2.1

Raise $x$ to the power of $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 (x \cdot x^{2}))}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 x^{1 + 2})}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.3.3

Add $1$ and $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (5 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.4

Multiply $- 4 \cdot 2 \cdot 5$ .

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Step 1.4.7.1.4.1

Multiply $- 4$ by $2$ .

$y = \frac{6 \pm \sqrt{36 - 8 \cdot (5 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.4.2

Multiply $- 8$ by $5$ .

$y = \frac{6 \pm \sqrt{36 - 40 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.5

Factor $4$ out of $36 - 40 x^{3}$ .

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Step 1.4.7.1.5.1

Factor $4$ out of $36$ .

$y = \frac{6 \pm \sqrt{4 (9) - 40 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.5.2

Factor $4$ out of $- 40 x^{3}$ .

$y = \frac{6 \pm \sqrt{4 (9) + 4 (- 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.5.3

Factor $4$ out of $4 (9) + 4 (- 10 x^{3})$ .

$y = \frac{6 \pm \sqrt{4 (9 - 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.6

Rewrite $4 (9 - 10 x^{3})$ as $2^{2} (3^{2} - 10 x^{3})$ .

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Step 1.4.7.1.6.1

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

$y = \frac{6 \pm \sqrt{2^{2} (9 - 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.6.2

Rewrite $9$ as $3^{2}$ .

$y = \frac{6 \pm \sqrt{2^{2} (3^{2} - 10 x^{3})}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.7

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{3^{2} - 10 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.7.1.8

Raise $3$ to the power of $2$ .

$y = \frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{2 \cdot (2 x^{2})}$

Step 1.4.7.2

Multiply $2$ by $2$ .

$y = \frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{2^{2} x^{2}}$

Step 1.4.7.3

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

$y = \frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{4 x^{2}}$

Step 1.4.7.4

Simplify $\frac{6 \pm 2 \sqrt{9 - 10 x^{3}}}{4 x^{2}}$ .

$y = \frac{3 \pm \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

Step 1.4.7.5

Change the $\pm$ to $-$ .

$y = \frac{3 - \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

Step 1.4.8

The final answer is the combination of both solutions.

$y = \frac{3 + \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

$y = \frac{3 - \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

$y = \frac{3 + \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

$y = \frac{3 - \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

$y = \frac{3 + \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

$y = \frac{3 - \sqrt{9 - 10 x^{3}}}{2 x^{2}}$

Step 2

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be $0$ or $1$ for each of its variables. In this case, the degree of the variable in the equation violates the linear equation definition, which means that the equation is not a linear equation.

Not Linear