Precalculus Examples

$x = 3 \sqrt{t}$ , $y = 9 - \sqrt{t}$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = 3 \sqrt{t}$

Step 2

Rewrite the equation as $3 \sqrt{t} = x$ .

$3 \sqrt{t} = x$

Step 3

Divide each term in $3 \sqrt{t} = x$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 \sqrt{t} = x$ by $3$ .

$\frac{3 \sqrt{t}}{3} = \frac{x}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 \sqrt{t}}{3} = \frac{x}{3}$

Step 3.2.1.2

Divide $\sqrt{t}$ by $1$ .

$\sqrt{t} = \frac{x}{3}$

Step 4

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

${\sqrt{t}}^{2} = {(\frac{x}{3})}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

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

${(t^{\frac{1}{2}})}^{2} = {(\frac{x}{3})}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(t^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Multiply the exponents in ${(t^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1.1

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

$t^{\frac{1}{2} \cdot 2} = {(\frac{x}{3})}^{2}$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

$t^{\frac{1}{2} \cdot 2} = {(\frac{x}{3})}^{2}$

Step 5.2.1.1.2.2

Rewrite the expression.

$t^{1} = {(\frac{x}{3})}^{2}$

Step 5.2.1.2

Simplify.

$t = {(\frac{x}{3})}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(\frac{x}{3})}^{2}$ .

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Step 5.3.1.1

Apply the product rule to $\frac{x}{3}$ .

$t = \frac{x^{2}}{3^{2}}$

Step 5.3.1.2

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

$t = \frac{x^{2}}{9}$

Step 6

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

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

Step 7

Simplify $9 - \sqrt{\frac{x^{2}}{9}}$ .

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Step 7.1

Simplify each term.

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Step 7.1.1

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

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

Step 7.1.2

Rewrite $\frac{x^{2}}{3^{2}}$ as ${(\frac{x}{3})}^{2}$ .

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

Step 7.1.3

Pull terms out from under the radical, assuming positive real numbers.

$y = 9 - \frac{x}{3}$

Step 7.2

Reorder $9$ and $- \frac{x}{3}$ .

$y = - \frac{x}{3} + 9$