Precalculus Examples

$x (t) = - 4 e^{t}$ , $y (t) = - \frac{3}{e^{t}}$

Step 1

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

$x = - 4 e^{t}$

Step 2

Rewrite the equation as $- 4 e^{t} = x$ .

$- 4 e^{t} = x$

Step 3

Divide each term in $- 4 e^{t} = x$ by $- 4$ and simplify.

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Step 3.1

Divide each term in $- 4 e^{t} = x$ by $- 4$ .

$\frac{- 4 e^{t}}{- 4} = \frac{x}{- 4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 4 e^{t}}{- 4} = \frac{x}{- 4}$

Step 3.2.1.2

Divide $e^{t}$ by $1$ .

$e^{t} = \frac{x}{- 4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$e^{t} = - \frac{x}{4}$

Step 4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{t}) = \ln (- \frac{x}{4})$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{t})$ by moving $t$ outside the logarithm.

$t \ln (e) = \ln (- \frac{x}{4})$

Step 5.2

The natural logarithm of $e$ is $1$ .

$t \cdot 1 = \ln (- \frac{x}{4})$

Step 5.3

Multiply $t$ by $1$ .

$t = \ln (- \frac{x}{4})$

Step 6

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

$y = - \frac{3}{e^{\ln (- \frac{x}{4})}}$

Step 7

Simplify $- \frac{3}{e^{\ln (- \frac{x}{4})}}$ .

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Step 7.1

Exponentiation and log are inverse functions.

$y = - \frac{3}{- \frac{x}{4}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$y = - (3 (- \frac{4}{x}))$

Step 7.3

Multiply $3 (- \frac{4}{x})$ .

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Step 7.3.1

Multiply $- 1$ by $3$ .

$y = - (- 3 \frac{4}{x})$

Step 7.3.2

Combine $- 3$ and $\frac{4}{x}$ .

$y = - \frac{- 3 \cdot 4}{x}$

Step 7.3.3

Multiply $- 3$ by $4$ .

$y = - \frac{- 12}{x}$

Step 7.4

Move the negative in front of the fraction.

$y = - - \frac{12}{x}$

Step 7.5

Multiply $- - \frac{12}{x}$ .

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Step 7.5.1

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{12}{x}$

Step 7.5.2

Multiply $\frac{12}{x}$ by $1$ .

$y = \frac{12}{x}$