Calculus Examples

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

Step 1

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

$x = \frac{1}{2} t + 3$

Step 2

Rewrite the equation as $\frac{1}{2} t + 3 = x$ .

$\frac{1}{2} t + 3 = x$

Step 3

Combine $\frac{1}{2}$ and $t$ .

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

Step 4

Subtract $3$ from both sides of the equation.

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

Step 5

Multiply both sides of the equation by $2$ .

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

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1

Cancel the common factor.

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

Step 6.1.1.2

Rewrite the expression.

$t = 2 (x - 3)$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $2 (x - 3)$ .

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Step 6.2.1.1

Apply the distributive property.

$t = 2 x + 2 \cdot - 3$

Step 6.2.1.2

Multiply $2$ by $- 3$ .

$t = 2 x - 6$

Step 7

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

$y = 2 (2 x - 6) - 1$

Step 8

Simplify $2 (2 x - 6) - 1$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

Apply the distributive property.

$y = 2 (2 x) + 2 \cdot - 6 - 1$

Step 8.1.2

Multiply $2$ by $2$ .

$y = 4 x + 2 \cdot - 6 - 1$

Step 8.1.3

Multiply $2$ by $- 6$ .

$y = 4 x - 12 - 1$

Step 8.2

Subtract $1$ from $- 12$ .

$y = 4 x - 13$