Algebra Examples

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

Step 1

Rewrite the equation as $\pm \sqrt{1 - 5 x} = y$ .

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Remove the plus-minus sign on $\pm {(1 - 5 x)}^{\frac{1}{2}}$ because it is raised to an even power.

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

Step 3.2.1.2

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

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Step 3.2.1.2.1

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

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

Step 3.2.1.2.2

Cancel the common factor of $2$ .

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Step 3.2.1.2.2.1

Cancel the common factor.

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

Step 3.2.1.2.2.2

Rewrite the expression.

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

Step 3.2.1.3

Simplify.

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

Step 4

Solve for $x$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 5$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Move the negative in front of the fraction.

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

Step 4.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 5

To rewrite as a function of $y$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $y$ is on the other side.

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