Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Subtract $1$ from both sides of the equation.

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

Step 4

Find the LCD of the terms in the equation.

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Step 4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$t, 1, 1$

Step 4.2

The LCM of one and any expression is the expression.

$t$

Step 5

Multiply each term in $\frac{1}{t} = x - 1$ by $t$ to eliminate the fractions.

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Step 5.1

Multiply each term in $\frac{1}{t} = x - 1$ by $t$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $t$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Rewrite the expression.

$1 = x t - t$

Step 6

Solve the equation.

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Step 6.1

Rewrite the equation as $x t - t = 1$ .

$x t - t = 1$

Step 6.2

Factor $t$ out of $x t - t$ .

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Step 6.2.1

Factor $t$ out of $x t$ .

$t x - t = 1$

Step 6.2.2

Factor $t$ out of $- t$ .

$t x + t \cdot - 1 = 1$

Step 6.2.3

Factor $t$ out of $t x + t \cdot - 1$ .

$t (x - 1) = 1$

Step 6.3

Divide each term in $t (x - 1) = 1$ by $x - 1$ and simplify.

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Step 6.3.1

Divide each term in $t (x - 1) = 1$ by $x - 1$ .

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

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $x - 1$ .

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Step 6.3.2.1.1

Cancel the common factor.

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

Step 6.3.2.1.2

Divide $t$ by $1$ .

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

Step 7

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

$y = 1 - \frac{1}{\frac{1}{x - 1}}$

Step 8

Simplify $1 - \frac{1}{\frac{1}{x - 1}}$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.1.2

Multiply $x - 1$ by $1$ .

$y = 1 - (x - 1)$

Step 8.1.3

Apply the distributive property.

$y = 1 - x - - 1$

Step 8.1.4

Multiply $- 1$ by $- 1$ .

$y = 1 - x + 1$

Step 8.2

Add $1$ and $1$ .

$y = - x + 2$