Precalculus Examples

$\frac{c - \sqrt{7}}{2 a - 5} = - \frac{1}{x}$

Step 1

Rewrite the equation as $- \frac{1}{x} = \frac{c - \sqrt{7}}{2 a - 5}$ .

$- \frac{1}{x} = \frac{c - \sqrt{7}}{2 a - 5}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$- (2 a - 5) = x (c - \sqrt{7})$

Step 3

Solve the equation for $x$ .

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Step 3.1

Rewrite the equation as $x (c - \sqrt{7}) = - (2 a - 5)$ .

$x (c - \sqrt{7}) = - (2 a - 5)$

Step 3.2

Simplify.

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Step 3.2.1

Apply the distributive property.

$x (c - \sqrt{7}) = - (2 a) - - 5$

Step 3.2.2

Multiply $2$ by $- 1$ .

$x (c - \sqrt{7}) = - 2 a - - 5$

Step 3.2.3

Multiply $- 1$ by $- 5$ .

$x (c - \sqrt{7}) = - 2 a + 5$

Step 3.3

Divide each term in $x (c - \sqrt{7}) = - 2 a + 5$ by $c - \sqrt{7}$ and simplify.

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Step 3.3.1

Divide each term in $x (c - \sqrt{7}) = - 2 a + 5$ by $c - \sqrt{7}$ .

$\frac{x (c - \sqrt{7})}{c - \sqrt{7}} = \frac{- 2 a}{c - \sqrt{7}} + \frac{5}{c - \sqrt{7}}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $c - \sqrt{7}$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{x (c - \sqrt{7})}{c - \sqrt{7}} = \frac{- 2 a}{c - \sqrt{7}} + \frac{5}{c - \sqrt{7}}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 a}{c - \sqrt{7}} + \frac{5}{c - \sqrt{7}}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Combine the numerators over the common denominator.

$x = \frac{- 2 a + 5}{c - \sqrt{7}}$

Step 3.3.3.2

Multiply $\frac{- 2 a + 5}{c - \sqrt{7}}$ by $\frac{c + \sqrt{7}}{c + \sqrt{7}}$ .

$x = \frac{- 2 a + 5}{c - \sqrt{7}} \cdot \frac{c + \sqrt{7}}{c + \sqrt{7}}$

Step 3.3.3.3

Multiply $\frac{- 2 a + 5}{c - \sqrt{7}}$ by $\frac{c + \sqrt{7}}{c + \sqrt{7}}$ .

$x = \frac{(- 2 a + 5) (c + \sqrt{7})}{(c - \sqrt{7}) (c + \sqrt{7})}$

Step 3.3.3.4

Expand the denominator using the FOIL method.

$x = \frac{(- 2 a + 5) (c + \sqrt{7})}{c^{2} + c \sqrt{7} - \sqrt{7} c - {\sqrt{7}}^{2}}$

Step 3.3.3.5

Simplify.

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Step 3.3.3.5.1

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.3.3.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \frac{(- 2 a + 5) (c + \sqrt{7})}{c^{2} - {(7^{\frac{1}{2}})}^{2}}$

Step 3.3.3.5.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{(- 2 a + 5) (c + \sqrt{7})}{c^{2} - 7^{\frac{1}{2} \cdot 2}}$

Step 3.3.3.5.1.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{(- 2 a + 5) (c + \sqrt{7})}{c^{2} - 7^{\frac{2}{2}}}$

Step 3.3.3.5.1.4

Cancel the common factor of $2$ .

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Step 3.3.3.5.1.4.1

Cancel the common factor.

$x = \frac{(- 2 a + 5) (c + \sqrt{7})}{c^{2} - 7^{\frac{2}{2}}}$

Step 3.3.3.5.1.4.2

Rewrite the expression.

$x = \frac{(- 2 a + 5) (c + \sqrt{7})}{c^{2} - 7^{1}}$

Step 3.3.3.5.1.5

Evaluate the exponent.

$x = \frac{(- 2 a + 5) (c + \sqrt{7})}{c^{2} - 1 \cdot 7}$

Step 3.3.3.5.2

Multiply $- 1$ by $7$ .

$x = \frac{(- 2 a + 5) (c + \sqrt{7})}{c^{2} - 7}$

Step 3.3.3.6

Factor $- 1$ out of $- 2 a$ .

$x = \frac{(- (2 a) + 5) (c + \sqrt{7})}{c^{2} - 7}$

Step 3.3.3.7

Rewrite $5$ as $- 1 (- 5)$ .

$x = \frac{(- (2 a) - 1 (- 5)) (c + \sqrt{7})}{c^{2} - 7}$

Step 3.3.3.8

Factor $- 1$ out of $- (2 a) - 1 (- 5)$ .

$x = \frac{- (2 a - 5) (c + \sqrt{7})}{c^{2} - 7}$

Step 3.3.3.9

Simplify the expression.

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Step 3.3.3.9.1

Rewrite $- (2 a - 5)$ as $- 1 (2 a - 5)$ .

$x = \frac{- 1 (2 a - 5) (c + \sqrt{7})}{c^{2} - 7}$

Step 3.3.3.9.2

Move the negative in front of the fraction.

$x = - \frac{(2 a - 5) (c + \sqrt{7})}{c^{2} - 7}$