Calculus Examples

$x = 3 t + 2$ , $y = 2 t + 3$

Step 1

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

$x = 3 t + 2$

Step 2

Rewrite the equation as $3 t + 2 = x$ .

$3 t + 2 = x$

Step 3

Subtract $2$ from both sides of the equation.

$3 t = x - 2$

Step 4

Divide each term in $3 t = x - 2$ by $3$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $3 t = x - 2$ by $3$ .

$\frac{3 t}{3} = \frac{x}{3} + \frac{- 2}{3}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{x}{3} + \frac{- 2}{3}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{x}{3} + \frac{- 2}{3}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

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

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Combine $2$ and $\frac{x}{3}$ .

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

Step 6.1.3

Multiply $2 (- \frac{2}{3})$ .

Tap for more steps...

Step 6.1.3.1

Multiply $- 1$ by $2$ .

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

Step 6.1.3.2

Combine $- 2$ and $\frac{2}{3}$ .

$y = \frac{2 x}{3} + \frac{- 2 \cdot 2}{3} + 3$

Step 6.1.3.3

Multiply $- 2$ by $2$ .

$y = \frac{2 x}{3} + \frac{- 4}{3} + 3$

Step 6.1.4

Move the negative in front of the fraction.

$y = \frac{2 x}{3} - \frac{4}{3} + 3$

Step 6.2

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

$y = \frac{2 x}{3} - \frac{4}{3} + 3 \cdot \frac{3}{3}$

Step 6.3

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

$y = \frac{2 x}{3} - \frac{4}{3} + \frac{3 \cdot 3}{3}$

Step 6.4

Combine the numerators over the common denominator.

$y = \frac{2 x}{3} + \frac{- 4 + 3 \cdot 3}{3}$

Step 6.5

Simplify the numerator.

Tap for more steps...

Step 6.5.1

Multiply $3$ by $3$ .

$y = \frac{2 x}{3} + \frac{- 4 + 9}{3}$

Step 6.5.2

Add $- 4$ and $9$ .

$y = \frac{2 x}{3} + \frac{5}{3}$