Precalculus Examples

$(\sqrt{8}, 2)$ , $(\sqrt{2}, - 2)$

Step 1

Find the slope of the line between $(\sqrt{8}, 2)$ and $(\sqrt{2}, - 2)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 1.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{- 2 - (2)}{\sqrt{2} - (\sqrt{8})}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Multiply $- 1$ by $2$ .

$m = \frac{- 2 - 2}{\sqrt{2} - \sqrt{8}}$

Step 1.4.1.2

Subtract $2$ from $- 2$ .

$m = \frac{- 4}{\sqrt{2} - \sqrt{8}}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.4.2.1.1

Factor $4$ out of $8$ .

$m = \frac{- 4}{\sqrt{2} - \sqrt{4 (2)}}$

Step 1.4.2.1.2

Rewrite $4$ as $2^{2}$ .

$m = \frac{- 4}{\sqrt{2} - \sqrt{2^{2} \cdot 2}}$

Step 1.4.2.2

Pull terms out from under the radical.

$m = \frac{- 4}{\sqrt{2} - (2 \sqrt{2})}$

Step 1.4.2.3

Multiply $2$ by $- 1$ .

$m = \frac{- 4}{\sqrt{2} - 2 \sqrt{2}}$

Step 1.4.2.4

Subtract $2 \sqrt{2}$ from $\sqrt{2}$ .

$m = \frac{- 4}{- \sqrt{2}}$

Step 1.4.3

Dividing two negative values results in a positive value.

$m = \frac{4}{\sqrt{2}}$

Step 1.4.4

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$m = \frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.4.5

Combine and simplify the denominator.

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Step 1.4.5.1

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$m = \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.4.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$m = \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.4.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$m = \frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.4.5.4

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

$m = \frac{4 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.4.5.5

Add $1$ and $1$ .

$m = \frac{4 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.4.5.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$m = \frac{4 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.4.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$m = \frac{4 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.4.5.6.3

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

$m = \frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.5.6.4

Cancel the common factor of $2$ .

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Step 1.4.5.6.4.1

Cancel the common factor.

$m = \frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.5.6.4.2

Rewrite the expression.

$m = \frac{4 \sqrt{2}}{2}$

Step 1.4.5.6.5

Evaluate the exponent.

$m = \frac{4 \sqrt{2}}{2}$

Step 1.4.6

Cancel the common factor of $4$ and $2$ .

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Step 1.4.6.1

Factor $2$ out of $4 \sqrt{2}$ .

$m = \frac{2 (2 \sqrt{2})}{2}$

Step 1.4.6.2

Cancel the common factors.

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Step 1.4.6.2.1

Factor $2$ out of $2$ .

$m = \frac{2 (2 \sqrt{2})}{2 (1)}$

Step 1.4.6.2.2

Cancel the common factor.

$m = \frac{2 (2 \sqrt{2})}{2 \cdot 1}$

Step 1.4.6.2.3

Rewrite the expression.

$m = \frac{2 \sqrt{2}}{1}$

Step 1.4.6.2.4

Divide $2 \sqrt{2}$ by $1$ .

$m = 2 \sqrt{2}$

Step 2

Use the slope $2 \sqrt{2}$ and a given point $(\sqrt{8}, 2)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = 2 \sqrt{2} \cdot (x - (\sqrt{8}))$

Step 3

Simplify the equation and keep it in point-slope form.

$y - 2 = 2 \sqrt{2} \cdot (x - 2 \sqrt{2})$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $2 \sqrt{2} \cdot (x - 2 \sqrt{2})$ .

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Step 4.1.1

Rewrite.

$y - 2 = 0 + 0 + 2 \sqrt{2} \cdot (x - 2 \sqrt{2})$

Step 4.1.2

Simplify by adding zeros.

$y - 2 = 2 \sqrt{2} \cdot (x - 2 \sqrt{2})$

Step 4.1.3

Apply the distributive property.

$y - 2 = 2 \sqrt{2} x + 2 \sqrt{2} (- 2 \sqrt{2})$

Step 4.1.4

Multiply $2 \sqrt{2} (- 2 \sqrt{2})$ .

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Step 4.1.4.1

Multiply $- 2$ by $2$ .

$y - 2 = 2 \sqrt{2} x - 4 \sqrt{2} \sqrt{2}$

Step 4.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$y - 2 = 2 \sqrt{2} x - 4 ({\sqrt{2}}^{1} \sqrt{2})$

Step 4.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$y - 2 = 2 \sqrt{2} x - 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 4.1.4.4

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

$y - 2 = 2 \sqrt{2} x - 4 {\sqrt{2}}^{1 + 1}$

Step 4.1.4.5

Add $1$ and $1$ .

$y - 2 = 2 \sqrt{2} x - 4 {\sqrt{2}}^{2}$

Step 4.1.5

Simplify each term.

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Step 4.1.5.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$y - 2 = 2 \sqrt{2} x - 4 {(2^{\frac{1}{2}})}^{2}$

Step 4.1.5.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.1.5.1.3

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

$y - 2 = 2 \sqrt{2} x - 4 \cdot 2^{\frac{2}{2}}$

Step 4.1.5.1.4

Cancel the common factor of $2$ .

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Step 4.1.5.1.4.1

Cancel the common factor.

$y - 2 = 2 \sqrt{2} x - 4 \cdot 2^{\frac{2}{2}}$

Step 4.1.5.1.4.2

Rewrite the expression.

$y - 2 = 2 \sqrt{2} x - 4 \cdot 2^{1}$

Step 4.1.5.1.5

Evaluate the exponent.

$y - 2 = 2 \sqrt{2} x - 4 \cdot 2$

Step 4.1.5.2

Multiply $- 4$ by $2$ .

$y - 2 = 2 \sqrt{2} x - 8$

Step 4.2

Move all terms not containing $y$ to the right side of the equation.

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Step 4.2.1

Add $2$ to both sides of the equation.

$y = 2 \sqrt{2} x - 8 + 2$

Step 4.2.2

Add $- 8$ and $2$ .

$y = 2 \sqrt{2} x - 6$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 2 \sqrt{2} x - 6$

Point-slope form:

$y - 2 = 2 \sqrt{2} \cdot (x - 2 \sqrt{2})$

Step 6