Precalculus Examples

$x = - \frac{1}{2} t + \frac{2}{3}$ , $y = t - \frac{3}{4}$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Subtract $\frac{2}{3}$ from both sides of the equation.

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

Step 5

Multiply both sides of the equation by $- 2$ .

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

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

Simplify $- 2 (- \frac{t}{2})$ .

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Step 6.1.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1.1

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

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

Step 6.1.1.1.2

Factor $2$ out of $- 2$ .

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

Step 6.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- t}{2} = - 2 (x - \frac{2}{3})$

Step 6.1.1.1.4

Rewrite the expression.

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

Step 6.1.1.2

Multiply.

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Step 6.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.1.2.2

Multiply $t$ by $1$ .

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

Step 6.2

Simplify the right side.

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Step 6.2.1

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

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Step 6.2.1.1

Apply the distributive property.

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

Step 6.2.1.2

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

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Step 6.2.1.2.1

Multiply $- 1$ by $- 2$ .

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

Step 6.2.1.2.2

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

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

Step 6.2.1.2.3

Multiply $2$ by $2$ .

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

Step 7

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

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

Step 8

Remove parentheses.

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

Step 9

Remove parentheses.

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

Step 10

Simplify $(- 2 x + \frac{4}{3}) - \frac{3}{4}$ .

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Step 10.1

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 10.2

To write $- \frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.3.1

Multiply $\frac{4}{3}$ by $\frac{4}{4}$ .

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

Step 10.3.2

Multiply $3$ by $4$ .

$y = - 2 x + \frac{4 \cdot 4}{12} - \frac{3}{4} \cdot \frac{3}{3}$

Step 10.3.3

Multiply $\frac{3}{4}$ by $\frac{3}{3}$ .

$y = - 2 x + \frac{4 \cdot 4}{12} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 10.3.4

Multiply $4$ by $3$ .

$y = - 2 x + \frac{4 \cdot 4}{12} - \frac{3 \cdot 3}{12}$

Step 10.4

Combine the numerators over the common denominator.

$y = - 2 x + \frac{4 \cdot 4 - 3 \cdot 3}{12}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Multiply $4$ by $4$ .

$y = - 2 x + \frac{16 - 3 \cdot 3}{12}$

Step 10.5.2

Multiply $- 3$ by $3$ .

$y = - 2 x + \frac{16 - 9}{12}$

Step 10.5.3

Subtract $9$ from $16$ .

$y = - 2 x + \frac{7}{12}$