Precalculus Examples

$x = - 2 t + 3$ , $y = 3 t + 1$

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

Simplify each term.

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Step 4.3.1.1

Move the negative in front of the fraction.

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

Step 4.3.1.2

Dividing two negative values results in a positive value.

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

Step 5

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

$y = 3 (- \frac{x}{2} + \frac{3}{2}) + 1$

Step 6

Simplify $3 (- \frac{x}{2} + \frac{3}{2}) + 1$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

$y = 3 (- \frac{x}{2}) + 3 (\frac{3}{2}) + 1$

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 1$ by $3$ .

$y = - 3 \frac{x}{2} + 3 (\frac{3}{2}) + 1$

Step 6.1.2.2

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

$y = \frac{- 3 x}{2} + 3 (\frac{3}{2}) + 1$

Step 6.1.3

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

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Step 6.1.3.1

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

$y = \frac{- 3 x}{2} + \frac{3 \cdot 3}{2} + 1$

Step 6.1.3.2

Multiply $3$ by $3$ .

$y = \frac{- 3 x}{2} + \frac{9}{2} + 1$

Step 6.1.4

Move the negative in front of the fraction.

$y = - \frac{3 x}{2} + \frac{9}{2} + 1$

Step 6.2

Simplify the expression.

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Step 6.2.1

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

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

Step 6.2.2

Combine the numerators over the common denominator.

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

Step 6.2.3

Add $9$ and $2$ .

$y = - \frac{3 x}{2} + \frac{11}{2}$