Calculus Examples

$y = - \sqrt{5 - x}$

Step 1

Rewrite the equation as $- \sqrt{5 - x} = y$ .

$- \sqrt{5 - x} = y$

Step 2

Divide each term in $- \sqrt{5 - x} = y$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \sqrt{5 - x} = y$ by $- 1$ .

$\frac{- \sqrt{5 - x}}{- 1} = \frac{y}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\sqrt{5 - x}}{1} = \frac{y}{- 1}$

Step 2.2.2

Divide $\sqrt{5 - x}$ by $1$ .

$\sqrt{5 - x} = \frac{y}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative one from the denominator of $\frac{y}{- 1}$ .

$\sqrt{5 - x} = - 1 \cdot y$

Step 2.3.2

Rewrite $- 1 \cdot y$ as $- y$ .

$\sqrt{5 - x} = - y$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 - x}$ as ${(5 - x)}^{\frac{1}{2}}$ .

${({(5 - x)}^{\frac{1}{2}})}^{2} = {(- y)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(5 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(5 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

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

${(5 - x)}^{\frac{1}{2} \cdot 2} = {(- y)}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(5 - x)}^{\frac{1}{2} \cdot 2} = {(- y)}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(5 - x)}^{1} = {(- y)}^{2}$

Step 4.2.1.2

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(- y)}^{2}$ .

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Step 4.3.1.1

Apply the product rule to $- y$ .

$5 - x = {(- 1)}^{2} y^{2}$

Step 4.3.1.2

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

$5 - x = 1 y^{2}$

Step 4.3.1.3

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

$5 - x = y^{2}$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $5$ from both sides of the equation.

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

Step 5.2

Divide each term in $- x = y^{2} - 5$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- x = y^{2} - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{y^{2}}{- 1} + \frac{- 5}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{y^{2}}{- 1} + \frac{- 5}{- 1}$

Step 5.2.2.2

Divide $x$ by $1$ .

$x = \frac{y^{2}}{- 1} + \frac{- 5}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative one from the denominator of $\frac{y^{2}}{- 1}$ .

$x = - 1 \cdot y^{2} + \frac{- 5}{- 1}$

Step 5.2.3.1.2

Rewrite $- 1 \cdot y^{2}$ as $- y^{2}$ .

$x = - y^{2} + \frac{- 5}{- 1}$

Step 5.2.3.1.3

Divide $- 5$ by $- 1$ .

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