Precalculus Examples

$x = 5 t - 6$ , $y = 25 t^{2}$

Step 1

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

$x = 5 t - 6$

Step 2

Rewrite the equation as $5 t - 6 = x$ .

$5 t - 6 = x$

Step 3

Add $6$ to both sides of the equation.

$5 t = x + 6$

Step 4

Divide each term in $5 t = x + 6$ by $5$ and simplify.

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Step 4.1

Divide each term in $5 t = x + 6$ by $5$ .

$\frac{5 t}{5} = \frac{x}{5} + \frac{6}{5}$

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.

$\frac{5 t}{5} = \frac{x}{5} + \frac{6}{5}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{x}{5} + \frac{6}{5}$

Step 5

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

$y = 25 {(\frac{x}{5} + \frac{6}{5})}^{2}$

Step 6

Simplify $25 {(\frac{x}{5} + \frac{6}{5})}^{2}$ .

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Step 6.1

Rewrite ${(\frac{x}{5} + \frac{6}{5})}^{2}$ as $(\frac{x}{5} + \frac{6}{5}) (\frac{x}{5} + \frac{6}{5})$ .

$y = 25 ((\frac{x}{5} + \frac{6}{5}) (\frac{x}{5} + \frac{6}{5}))$

Step 6.2

Expand $(\frac{x}{5} + \frac{6}{5}) (\frac{x}{5} + \frac{6}{5})$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

$y = 25 (\frac{x}{5} (\frac{x}{5} + \frac{6}{5}) + \frac{6}{5} (\frac{x}{5} + \frac{6}{5}))$

Step 6.2.2

Apply the distributive property.

$y = 25 (\frac{x}{5} \cdot \frac{x}{5} + \frac{x}{5} \cdot \frac{6}{5} + \frac{6}{5} (\frac{x}{5} + \frac{6}{5}))$

Step 6.2.3

Apply the distributive property.

$y = 25 (\frac{x}{5} \cdot \frac{x}{5} + \frac{x}{5} \cdot \frac{6}{5} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

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

$y = 25 (\frac{x \cdot x}{5 \cdot 5} + \frac{x}{5} \cdot \frac{6}{5} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.1.2

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

$y = 25 (\frac{x^{1} x}{5 \cdot 5} + \frac{x}{5} \cdot \frac{6}{5} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.1.3

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

$y = 25 (\frac{x^{1} x^{1}}{5 \cdot 5} + \frac{x}{5} \cdot \frac{6}{5} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.1.4

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

$y = 25 (\frac{x^{1 + 1}}{5 \cdot 5} + \frac{x}{5} \cdot \frac{6}{5} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.1.5

Add $1$ and $1$ .

$y = 25 (\frac{x^{2}}{5 \cdot 5} + \frac{x}{5} \cdot \frac{6}{5} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.1.6

Multiply $5$ by $5$ .

$y = 25 (\frac{x^{2}}{25} + \frac{x}{5} \cdot \frac{6}{5} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.2

Multiply $\frac{x}{5} \cdot \frac{6}{5}$ .

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Step 6.3.1.2.1

Multiply $\frac{x}{5}$ by $\frac{6}{5}$ .

$y = 25 (\frac{x^{2}}{25} + \frac{x \cdot 6}{5 \cdot 5} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.2.2

Multiply $5$ by $5$ .

$y = 25 (\frac{x^{2}}{25} + \frac{x \cdot 6}{25} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.3

Move $6$ to the left of $x$ .

$y = 25 (\frac{x^{2}}{25} + \frac{6 \cdot x}{25} + \frac{6}{5} \cdot \frac{x}{5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.4

Multiply $\frac{6}{5} \cdot \frac{x}{5}$ .

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Step 6.3.1.4.1

Multiply $\frac{6}{5}$ by $\frac{x}{5}$ .

$y = 25 (\frac{x^{2}}{25} + \frac{6 x}{25} + \frac{6 x}{5 \cdot 5} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.4.2

Multiply $5$ by $5$ .

$y = 25 (\frac{x^{2}}{25} + \frac{6 x}{25} + \frac{6 x}{25} + \frac{6}{5} \cdot \frac{6}{5})$

Step 6.3.1.5

Multiply $\frac{6}{5} \cdot \frac{6}{5}$ .

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Step 6.3.1.5.1

Multiply $\frac{6}{5}$ by $\frac{6}{5}$ .

$y = 25 (\frac{x^{2}}{25} + \frac{6 x}{25} + \frac{6 x}{25} + \frac{6 \cdot 6}{5 \cdot 5})$

Step 6.3.1.5.2

Multiply $6$ by $6$ .

$y = 25 (\frac{x^{2}}{25} + \frac{6 x}{25} + \frac{6 x}{25} + \frac{36}{5 \cdot 5})$

Step 6.3.1.5.3

Multiply $5$ by $5$ .

$y = 25 (\frac{x^{2}}{25} + \frac{6 x}{25} + \frac{6 x}{25} + \frac{36}{25})$

Step 6.3.2

Add $\frac{6 x}{25}$ and $\frac{6 x}{25}$ .

$y = 25 (\frac{x^{2}}{25} + 2 \frac{6 x}{25} + \frac{36}{25})$

Step 6.4

Multiply $2 \frac{6 x}{25}$ .

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Step 6.4.1

Combine $2$ and $\frac{6 x}{25}$ .

$y = 25 (\frac{x^{2}}{25} + \frac{2 (6 x)}{25} + \frac{36}{25})$

Step 6.4.2

Multiply $6$ by $2$ .

$y = 25 (\frac{x^{2}}{25} + \frac{12 x}{25} + \frac{36}{25})$

Step 6.5

Apply the distributive property.

$y = 25 \frac{x^{2}}{25} + 25 \frac{12 x}{25} + 25 (\frac{36}{25})$

Step 6.6

Simplify.

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Step 6.6.1

Cancel the common factor of $25$ .

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Step 6.6.1.1

Cancel the common factor.

$y = 25 \frac{x^{2}}{25} + 25 \frac{12 x}{25} + 25 (\frac{36}{25})$

Step 6.6.1.2

Rewrite the expression.

$y = x^{2} + 25 \frac{12 x}{25} + 25 (\frac{36}{25})$

Step 6.6.2

Cancel the common factor of $25$ .

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Step 6.6.2.1

Cancel the common factor.

$y = x^{2} + 25 \frac{12 x}{25} + 25 (\frac{36}{25})$

Step 6.6.2.2

Rewrite the expression.

$y = x^{2} + 12 x + 25 (\frac{36}{25})$

Step 6.6.3

Cancel the common factor of $25$ .

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Step 6.6.3.1

Cancel the common factor.

$y = x^{2} + 12 x + 25 (\frac{36}{25})$

Step 6.6.3.2

Rewrite the expression.

$y = x^{2} + 12 x + 36$