Precalculus Examples

$x = 10 + 5 t$ , $y = 50 - 5 t^{2}$

Step 1

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

$x = 10 + 5 t$

Step 2

Rewrite the equation as $10 + 5 t = x$ .

$10 + 5 t = x$

Step 3

Subtract $10$ from both sides of the equation.

$5 t = x - 10$

Step 4

Divide each term in $5 t = x - 10$ by $5$ and simplify.

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Step 4.1

Divide each term in $5 t = x - 10$ by $5$ .

$\frac{5 t}{5} = \frac{x}{5} + \frac{- 10}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{5 t}{5} = \frac{x}{5} + \frac{- 10}{5}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{x}{5} + \frac{- 10}{5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 10$ by $5$ .

$t = \frac{x}{5} - 2$

Step 5

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

$y = 50 - 5 {(\frac{x}{5} - 2)}^{2}$

Step 6

Simplify $50 - 5 {(\frac{x}{5} - 2)}^{2}$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite ${(\frac{x}{5} - 2)}^{2}$ as $(\frac{x}{5} - 2) (\frac{x}{5} - 2)$ .

$y = 50 - 5 ((\frac{x}{5} - 2) (\frac{x}{5} - 2))$

Step 6.1.2

Expand $(\frac{x}{5} - 2) (\frac{x}{5} - 2)$ using the FOIL Method.

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Step 6.1.2.1

Apply the distributive property.

$y = 50 - 5 (\frac{x}{5} (\frac{x}{5} - 2) - 2 (\frac{x}{5} - 2))$

Step 6.1.2.2

Apply the distributive property.

$y = 50 - 5 (\frac{x}{5} \cdot \frac{x}{5} + \frac{x}{5} \cdot - 2 - 2 (\frac{x}{5} - 2))$

Step 6.1.2.3

Apply the distributive property.

$y = 50 - 5 (\frac{x}{5} \cdot \frac{x}{5} + \frac{x}{5} \cdot - 2 - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3

Simplify and combine like terms.

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Step 6.1.3.1

Simplify each term.

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Step 6.1.3.1.1

Multiply $\frac{x}{5} \cdot \frac{x}{5}$ .

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Step 6.1.3.1.1.1

Multiply $\frac{x}{5}$ by $\frac{x}{5}$ .

$y = 50 - 5 (\frac{x \cdot x}{5 \cdot 5} + \frac{x}{5} \cdot - 2 - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.1.2

Raise $x$ to the power of $1$ .

$y = 50 - 5 (\frac{x^{1} x}{5 \cdot 5} + \frac{x}{5} \cdot - 2 - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.1.3

Raise $x$ to the power of $1$ .

$y = 50 - 5 (\frac{x^{1} x^{1}}{5 \cdot 5} + \frac{x}{5} \cdot - 2 - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 50 - 5 (\frac{x^{1 + 1}}{5 \cdot 5} + \frac{x}{5} \cdot - 2 - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.1.5

Add $1$ and $1$ .

$y = 50 - 5 (\frac{x^{2}}{5 \cdot 5} + \frac{x}{5} \cdot - 2 - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.1.6

Multiply $5$ by $5$ .

$y = 50 - 5 (\frac{x^{2}}{25} + \frac{x}{5} \cdot - 2 - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.2

Combine $\frac{x}{5}$ and $- 2$ .

$y = 50 - 5 (\frac{x^{2}}{25} + \frac{x \cdot - 2}{5} - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.3

Move $- 2$ to the left of $x$ .

$y = 50 - 5 (\frac{x^{2}}{25} + \frac{- 2 \cdot x}{5} - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.4

Move the negative in front of the fraction.

$y = 50 - 5 (\frac{x^{2}}{25} - \frac{2 \cdot x}{5} - 2 \frac{x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.5

Combine $- 2$ and $\frac{x}{5}$ .

$y = 50 - 5 (\frac{x^{2}}{25} - \frac{2 x}{5} + \frac{- 2 x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.6

Move the negative in front of the fraction.

$y = 50 - 5 (\frac{x^{2}}{25} - \frac{2 x}{5} - \frac{2 x}{5} - 2 \cdot - 2)$

Step 6.1.3.1.7

Multiply $- 2$ by $- 2$ .

$y = 50 - 5 (\frac{x^{2}}{25} - \frac{2 x}{5} - \frac{2 x}{5} + 4)$

Step 6.1.3.2

Subtract $\frac{2 x}{5}$ from $- \frac{2 x}{5}$ .

$y = 50 - 5 (\frac{x^{2}}{25} - 2 \frac{2 x}{5} + 4)$

Step 6.1.4

Simplify each term.

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Step 6.1.4.1

Multiply $- 2 \frac{2 x}{5}$ .

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Step 6.1.4.1.1

Combine $- 2$ and $\frac{2 x}{5}$ .

$y = 50 - 5 (\frac{x^{2}}{25} + \frac{- 2 (2 x)}{5} + 4)$

Step 6.1.4.1.2

Multiply $2$ by $- 2$ .

$y = 50 - 5 (\frac{x^{2}}{25} + \frac{- 4 x}{5} + 4)$

Step 6.1.4.2

Move the negative in front of the fraction.

$y = 50 - 5 (\frac{x^{2}}{25} - \frac{4 x}{5} + 4)$

Step 6.1.5

Apply the distributive property.

$y = 50 - 5 \frac{x^{2}}{25} - 5 (- \frac{4 x}{5}) - 5 \cdot 4$

Step 6.1.6

Simplify.

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Step 6.1.6.1

Cancel the common factor of $5$ .

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Step 6.1.6.1.1

Factor $5$ out of $- 5$ .

$y = 50 + 5 (- 1) \frac{x^{2}}{25} - 5 (- \frac{4 x}{5}) - 5 \cdot 4$

Step 6.1.6.1.2

Factor $5$ out of $25$ .

$y = 50 + 5 \cdot - 1 \frac{x^{2}}{5 \cdot 5} - 5 (- \frac{4 x}{5}) - 5 \cdot 4$

Step 6.1.6.1.3

Cancel the common factor.

$y = 50 + 5 \cdot - 1 \frac{x^{2}}{5 \cdot 5} - 5 (- \frac{4 x}{5}) - 5 \cdot 4$

Step 6.1.6.1.4

Rewrite the expression.

$y = 50 - 1 \frac{x^{2}}{5} - 5 (- \frac{4 x}{5}) - 5 \cdot 4$

Step 6.1.6.2

Cancel the common factor of $5$ .

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Step 6.1.6.2.1

Move the leading negative in $- \frac{4 x}{5}$ into the numerator.

$y = 50 - 1 \frac{x^{2}}{5} - 5 \frac{- 4 x}{5} - 5 \cdot 4$

Step 6.1.6.2.2

Factor $5$ out of $- 5$ .

$y = 50 - 1 \frac{x^{2}}{5} + 5 (- 1) \frac{- 4 x}{5} - 5 \cdot 4$

Step 6.1.6.2.3

Cancel the common factor.

$y = 50 - 1 \frac{x^{2}}{5} + 5 \cdot - 1 \frac{- 4 x}{5} - 5 \cdot 4$

Step 6.1.6.2.4

Rewrite the expression.

$y = 50 - 1 \frac{x^{2}}{5} - 1 (- 4 x) - 5 \cdot 4$

Step 6.1.6.3

Multiply $- 4$ by $- 1$ .

$y = 50 - 1 \frac{x^{2}}{5} + 4 x - 5 \cdot 4$

Step 6.1.6.4

Multiply $- 5$ by $4$ .

$y = 50 - 1 \frac{x^{2}}{5} + 4 x - 20$

Step 6.1.7

Rewrite $- 1 \frac{x^{2}}{5}$ as $- \frac{x^{2}}{5}$ .

$y = 50 - \frac{x^{2}}{5} + 4 x - 20$

Step 6.2

Subtract $20$ from $50$ .

$y = - \frac{x^{2}}{5} + 4 x + 30$