Finite Math Examples

$- 7 y^{2} = - 6 + \frac{1}{4} x$

Step 1

Divide each term in $- 7 y^{2} = - 6 + \frac{1}{4} x$ by $- 7$ and simplify.

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Step 1.1

Divide each term in $- 7 y^{2} = - 6 + \frac{1}{4} x$ by $- 7$ .

$\frac{- 7 y^{2}}{- 7} = \frac{- 6}{- 7} + \frac{\frac{1}{4} x}{- 7}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 7$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{- 7 y^{2}}{- 7} = \frac{- 6}{- 7} + \frac{\frac{1}{4} x}{- 7}$

Step 1.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 6}{- 7} + \frac{\frac{1}{4} x}{- 7}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Dividing two negative values results in a positive value.

$y^{2} = \frac{6}{7} + \frac{\frac{1}{4} x}{- 7}$

Step 1.3.1.2

Combine $\frac{1}{4}$ and $x$ .

$y^{2} = \frac{6}{7} + \frac{\frac{x}{4}}{- 7}$

Step 1.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y^{2} = \frac{6}{7} + \frac{x}{4} \cdot \frac{1}{- 7}$

Step 1.3.1.4

Move the negative in front of the fraction.

$y^{2} = \frac{6}{7} + \frac{x}{4} (- \frac{1}{7})$

Step 1.3.1.5

Multiply $\frac{x}{4} (- \frac{1}{7})$ .

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Step 1.3.1.5.1

Multiply $\frac{x}{4}$ by $\frac{1}{7}$ .

$y^{2} = \frac{6}{7} - \frac{x}{4 \cdot 7}$

Step 1.3.1.5.2

Multiply $4$ by $7$ .

$y^{2} = \frac{6}{7} - \frac{x}{28}$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{6}{7} - \frac{x}{28}}$

Step 3

Simplify $\pm \sqrt{\frac{6}{7} - \frac{x}{28}}$ .

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Step 3.1

To write $\frac{6}{7}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \pm \sqrt{\frac{6}{7} \cdot \frac{4}{4} - \frac{x}{28}}$

Step 3.2

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.1

Multiply $\frac{6}{7}$ by $\frac{4}{4}$ .

$y = \pm \sqrt{\frac{6 \cdot 4}{7 \cdot 4} - \frac{x}{28}}$

Step 3.2.2

Multiply $7$ by $4$ .

$y = \pm \sqrt{\frac{6 \cdot 4}{28} - \frac{x}{28}}$

Step 3.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{6 \cdot 4 - x}{28}}$

Step 3.4

Multiply $6$ by $4$ .

$y = \pm \sqrt{\frac{24 - x}{28}}$

Step 3.5

Rewrite $\frac{24 - x}{28}$ as ${(\frac{1}{2})}^{2} \frac{24 - x}{7}$ .

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Step 3.5.1

Factor the perfect power $1^{2}$ out of $24 - x$ .

$y = \pm \sqrt{\frac{1^{2} (24 - x)}{28}}$

Step 3.5.2

Factor the perfect power $2^{2}$ out of $28$ .

$y = \pm \sqrt{\frac{1^{2} (24 - x)}{2^{2} \cdot 7}}$

Step 3.5.3

Rearrange the fraction $\frac{1^{2} (24 - x)}{2^{2} \cdot 7}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} \frac{24 - x}{7}}$

Step 3.6

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt{\frac{24 - x}{7}}$

Step 3.7

Rewrite $\sqrt{\frac{24 - x}{7}}$ as $\frac{\sqrt{24 - x}}{\sqrt{7}}$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x}}{\sqrt{7}}$

Step 3.8

Multiply $\frac{\sqrt{24 - x}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{1}{2} (\frac{\sqrt{24 - x}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 3.9

Combine and simplify the denominator.

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Step 3.9.1

Multiply $\frac{\sqrt{24 - x}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.9.2

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 3.9.3

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 3.9.4

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

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 3.9.5

Add $1$ and $1$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 3.9.6

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

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Step 3.9.6.1

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

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 3.9.6.2

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

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 3.9.6.3

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

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.9.6.4

Cancel the common factor of $2$ .

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Step 3.9.6.4.1

Cancel the common factor.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.9.6.4.2

Rewrite the expression.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{7^{1}}$

Step 3.9.6.5

Evaluate the exponent.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{24 - x} \sqrt{7}}{7}$

Step 3.10

Combine using the product rule for radicals.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{(24 - x) \cdot 7}}{7}$

Step 3.11

Multiply $\frac{1}{2} \cdot \frac{\sqrt{(24 - x) \cdot 7}}{7}$ .

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Step 3.11.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{(24 - x) \cdot 7}}{7}$ .

$y = \pm \frac{\sqrt{(24 - x) \cdot 7}}{2 \cdot 7}$

Step 3.11.2

Multiply $2$ by $7$ .

$y = \pm \frac{\sqrt{(24 - x) \cdot 7}}{14}$

Step 3.12

Reorder factors in $\pm \frac{\sqrt{(24 - x) \cdot 7}}{14}$ .

$y = \pm \frac{\sqrt{7 (24 - x)}}{14}$

Step 4

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

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Step 4.1

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

$y = \frac{\sqrt{7 (24 - x)}}{14}$

Step 4.2

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

$y = - \frac{\sqrt{7 (24 - x)}}{14}$

Step 4.3

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

$y = \frac{\sqrt{7 (24 - x)}}{14}$

$y = - \frac{\sqrt{7 (24 - x)}}{14}$

$y = \frac{\sqrt{7 (24 - x)}}{14}$

$y = - \frac{\sqrt{7 (24 - x)}}{14}$