Calculus Examples

$x = 2 + 4 t$ and $y = - 1 + 6 t$

Step 1

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

$x = 2 + 4 t$

Step 2

Rewrite the equation as $2 + 4 t = x$ .

$2 + 4 t = x$

Step 3

Subtract $2$ from both sides of the equation.

$4 t = x - 2$

Step 4

Divide each term in $4 t = x - 2$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 t = x - 2$ by $4$ .

$\frac{4 t}{4} = \frac{x}{4} + \frac{- 2}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 t}{4} = \frac{x}{4} + \frac{- 2}{4}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{x}{4} + \frac{- 2}{4}$

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

Cancel the common factor of $- 2$ and $4$ .

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Step 4.3.1.1.1

Factor $2$ out of $- 2$ .

$t = \frac{x}{4} + \frac{2 (- 1)}{4}$

Step 4.3.1.1.2

Cancel the common factors.

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Step 4.3.1.1.2.1

Factor $2$ out of $4$ .

$t = \frac{x}{4} + \frac{2 \cdot - 1}{2 \cdot 2}$

Step 4.3.1.1.2.2

Cancel the common factor.

$t = \frac{x}{4} + \frac{2 \cdot - 1}{2 \cdot 2}$

Step 4.3.1.1.2.3

Rewrite the expression.

$t = \frac{x}{4} + \frac{- 1}{2}$

Step 4.3.1.2

Move the negative in front of the fraction.

$t = \frac{x}{4} - \frac{1}{2}$

Step 5

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

$y = - 1 + 6 (\frac{x}{4} - \frac{1}{2})$

Step 6

Simplify $- 1 + 6 (\frac{x}{4} - \frac{1}{2})$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

$y = - 1 + 6 \frac{x}{4} + 6 (- \frac{1}{2})$

Step 6.1.2

Cancel the common factor of $2$ .

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Step 6.1.2.1

Factor $2$ out of $6$ .

$y = - 1 + 2 (3) \frac{x}{4} + 6 (- \frac{1}{2})$

Step 6.1.2.2

Factor $2$ out of $4$ .

$y = - 1 + 2 \cdot 3 \frac{x}{2 \cdot 2} + 6 (- \frac{1}{2})$

Step 6.1.2.3

Cancel the common factor.

$y = - 1 + 2 \cdot 3 \frac{x}{2 \cdot 2} + 6 (- \frac{1}{2})$

Step 6.1.2.4

Rewrite the expression.

$y = - 1 + 3 \frac{x}{2} + 6 (- \frac{1}{2})$

Step 6.1.3

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

$y = - 1 + \frac{3 x}{2} + 6 (- \frac{1}{2})$

Step 6.1.4

Cancel the common factor of $2$ .

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Step 6.1.4.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y = - 1 + \frac{3 x}{2} + 6 (\frac{- 1}{2})$

Step 6.1.4.2

Factor $2$ out of $6$ .

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

Step 6.1.4.3

Cancel the common factor.

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

Step 6.1.4.4

Rewrite the expression.

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

Step 6.1.5

Multiply $3$ by $- 1$ .

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

Step 6.2

Subtract $3$ from $- 1$ .

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