Precalculus Examples

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

Step 1

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

$x = 2 t + 3$

Step 2

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

$2 t + 3 = x$

Step 3

Subtract $3$ from both sides of the equation.

$2 t = x - 3$

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $t$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

Remove parentheses.

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

Step 7

Remove parentheses.

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

Step 8

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

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Step 8.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.2

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

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

Step 8.3

Combine the numerators over the common denominator.

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

Step 8.4

Simplify the numerator.

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Step 8.4.1

Multiply $- 2$ by $2$ .

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

Step 8.4.2

Subtract $4$ from $- 3$ .

$y = \frac{x}{2} + \frac{- 7}{2}$

Step 8.5

Move the negative in front of the fraction.

$y = \frac{x}{2} - \frac{7}{2}$