Precalculus Examples

$x (t) = 3 t + 4$ $y (t) = 5 - t$

Step 1

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

$x = 3 t + 4$

Step 2

Rewrite the equation as $3 t + 4 = x$ .

$3 t + 4 = x$

Step 3

Subtract $4$ from both sides of the equation.

$3 t = x - 4$

Step 4

Divide each term in $3 t = x - 4$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 t = x - 4$ by $3$ .

$\frac{3 t}{3} = \frac{x}{3} + \frac{- 4}{3}$

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.

$\frac{3 t}{3} = \frac{x}{3} + \frac{- 4}{3}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{x}{3} + \frac{- 4}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

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

Step 5

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

$y = 5 - (\frac{x}{3} - \frac{4}{3})$

Step 6

Simplify $5 - (\frac{x}{3} - \frac{4}{3})$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

$y = 5 - \frac{x}{3} - - \frac{4}{3}$

Step 6.1.2

Multiply $- - \frac{4}{3}$ .

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Step 6.1.2.1

Multiply $- 1$ by $- 1$ .

$y = 5 - \frac{x}{3} + 1 (\frac{4}{3})$

Step 6.1.2.2

Multiply $\frac{4}{3}$ by $1$ .

$y = 5 - \frac{x}{3} + \frac{4}{3}$

Step 6.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{x}{3} + 5 \cdot \frac{3}{3} + \frac{4}{3}$

Step 6.3

Combine $5$ and $\frac{3}{3}$ .

$y = - \frac{x}{3} + \frac{5 \cdot 3}{3} + \frac{4}{3}$

Step 6.4

Combine the numerators over the common denominator.

$y = - \frac{x}{3} + \frac{5 \cdot 3 + 4}{3}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $5$ by $3$ .

$y = - \frac{x}{3} + \frac{15 + 4}{3}$

Step 6.5.2

Add $15$ and $4$ .

$y = - \frac{x}{3} + \frac{19}{3}$