Precalculus Examples

$x (t) = 1 + 3 t$ , $y (t) = \ln (t + 1)$

Step 1

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

$x = 1 + 3 t$

Step 2

Rewrite the equation as $1 + 3 t = x$ .

$1 + 3 t = x$

Step 3

Subtract $1$ from both sides of the equation.

$3 t = x - 1$

Step 4

Divide each term in $3 t = x - 1$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 t = x - 1$ by $3$ .

$\frac{3 t}{3} = \frac{x}{3} + \frac{- 1}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{x}{3} + \frac{- 1}{3}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{x}{3} + \frac{- 1}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$t = \frac{x}{3} - \frac{1}{3}$

Step 5

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

$y = \ln ((\frac{x}{3} - \frac{1}{3}) + 1)$

Step 6

Remove parentheses.

$y = \ln (\frac{x}{3} - \frac{1}{3} + 1)$

Step 7

Remove parentheses.

$y = \ln ((\frac{x}{3} - \frac{1}{3}) + 1)$

Step 8

Simplify $\ln ((\frac{x}{3} - \frac{1}{3}) + 1)$ .

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Step 8.1

Write $1$ as a fraction with a common denominator.

$y = \ln (\frac{x}{3} - \frac{1}{3} + \frac{3}{3})$

Step 8.2

Combine the numerators over the common denominator.

$y = \ln (\frac{x}{3} + \frac{- 1 + 3}{3})$

Step 8.3

Add $- 1$ and $3$ .

$y = \ln (\frac{x}{3} + \frac{2}{3})$