Precalculus Examples

$(- \frac{1}{2}, \sqrt{\frac{3}{2}})$ , $(0, 0)$

Step 1

Find the slope of the line between $(- \frac{1}{2}, \sqrt{\frac{3}{2}})$ and $(0, 0)$ 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.

$m = \frac{0 - (\sqrt{\frac{3}{2}})}{0 - (- \frac{1}{2})}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$m = \frac{0 - \frac{\sqrt{3}}{\sqrt{2}}}{0 + \frac{1}{2}}$

Step 1.4.1.2

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$m = \frac{0 - (\frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}{0 + \frac{1}{2}}$

Step 1.4.1.3

Combine and simplify the denominator.

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Step 1.4.1.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}}{0 + \frac{1}{2}}$

Step 1.4.1.3.2

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

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}}{0 + \frac{1}{2}}$

Step 1.4.1.3.3

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

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}}{0 + \frac{1}{2}}$

Step 1.4.1.3.4

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

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}}{0 + \frac{1}{2}}$

Step 1.4.1.3.5

Add $1$ and $1$ .

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}}{0 + \frac{1}{2}}$

Step 1.4.1.3.6

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

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Step 1.4.1.3.6.1

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

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}{0 + \frac{1}{2}}$

Step 1.4.1.3.6.2

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

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}{0 + \frac{1}{2}}$

Step 1.4.1.3.6.3

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

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}}{0 + \frac{1}{2}}$

Step 1.4.1.3.6.4

Cancel the common factor of $2$ .

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Step 1.4.1.3.6.4.1

Cancel the common factor.

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}}{0 + \frac{1}{2}}$

Step 1.4.1.3.6.4.2

Rewrite the expression.

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{2}}{0 + \frac{1}{2}}$

Step 1.4.1.3.6.5

Evaluate the exponent.

$m = \frac{0 - \frac{\sqrt{3} \sqrt{2}}{2}}{0 + \frac{1}{2}}$

Step 1.4.1.4

Simplify the numerator.

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Step 1.4.1.4.1

Combine using the product rule for radicals.

$m = \frac{0 - \frac{\sqrt{3 \cdot 2}}{2}}{0 + \frac{1}{2}}$

Step 1.4.1.4.2

Multiply $3$ by $2$ .

$m = \frac{0 - \frac{\sqrt{6}}{2}}{0 + \frac{1}{2}}$

Step 1.4.1.5

Subtract $\frac{\sqrt{6}}{2}$ from $0$ .

$m = \frac{- \frac{\sqrt{6}}{2}}{0 + \frac{1}{2}}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- - \frac{1}{2}$ .

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Step 1.4.2.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{- \frac{\sqrt{6}}{2}}{0 + 1 (\frac{1}{2})}$

Step 1.4.2.1.2

Multiply $\frac{1}{2}$ by $1$ .

$m = \frac{- \frac{\sqrt{6}}{2}}{0 + \frac{1}{2}}$

Step 1.4.2.2

Add $0$ and $\frac{1}{2}$ .

$m = \frac{- \frac{\sqrt{6}}{2}}{\frac{1}{2}}$

Step 1.4.3

Multiply the numerator by the reciprocal of the denominator.

$m = - \frac{\sqrt{6}}{2} \cdot 2$

Step 1.4.4

Cancel the common factor of $2$ .

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Step 1.4.4.1

Move the leading negative in $- \frac{\sqrt{6}}{2}$ into the numerator.

$m = \frac{- \sqrt{6}}{2} \cdot 2$

Step 1.4.4.2

Cancel the common factor.

$m = \frac{- \sqrt{6}}{2} \cdot 2$

Step 1.4.4.3

Rewrite the expression.

$m = - \sqrt{6}$

Step 2

Use the slope $- \sqrt{6}$ and a given point $(- \frac{1}{2}, \sqrt{\frac{3}{2}})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\sqrt{\frac{3}{2}}) = - \sqrt{6} \cdot (x - (- \frac{1}{2}))$

Step 3

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

$y - \frac{\sqrt{6}}{2} = - \sqrt{6} \cdot (x + \frac{1}{2})$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $- \sqrt{6} \cdot (x + \frac{1}{2})$ .

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Step 4.1.1

Rewrite.

$y - \frac{\sqrt{6}}{2} = 0 + 0 - \sqrt{6} \cdot (x + \frac{1}{2})$

Step 4.1.2

Simplify by adding zeros.

$y - \frac{\sqrt{6}}{2} = - \sqrt{6} \cdot (x + \frac{1}{2})$

Step 4.1.3

Apply the distributive property.

$y - \frac{\sqrt{6}}{2} = - \sqrt{6} x - \sqrt{6} \frac{1}{2}$

Step 4.1.4

Combine $\frac{1}{2}$ and $\sqrt{6}$ .

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

Step 4.1.5

Reorder the factors of $- \sqrt{6} x$ .

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

Step 4.1.6

To write $- x \sqrt{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.1.7

Combine $- x \sqrt{6}$ and $\frac{2}{2}$ .

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

Step 4.1.8

Combine the numerators over the common denominator.

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

Step 4.1.9

Multiply $2$ by $- 1$ .

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

Step 4.1.10

Factor $- 1$ out of $- 2 x \sqrt{6}$ .

$y - \frac{\sqrt{6}}{2} = \frac{- (2 x \sqrt{6}) - \sqrt{6}}{2}$

Step 4.1.11

Factor $- 1$ out of $- \sqrt{6}$ .

$y - \frac{\sqrt{6}}{2} = \frac{- (2 x \sqrt{6}) - (\sqrt{6})}{2}$

Step 4.1.12

Factor $- 1$ out of $- (2 x \sqrt{6}) - (\sqrt{6})$ .

$y - \frac{\sqrt{6}}{2} = \frac{- (2 x \sqrt{6} + \sqrt{6})}{2}$

Step 4.1.13

Simplify the expression.

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Step 4.1.13.1

Rewrite $- (2 x \sqrt{6} + \sqrt{6})$ as $- 1 (2 x \sqrt{6} + \sqrt{6})$ .

$y - \frac{\sqrt{6}}{2} = \frac{- 1 (2 x \sqrt{6} + \sqrt{6})}{2}$

Step 4.1.13.2

Move the negative in front of the fraction.

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

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 $\frac{\sqrt{6}}{2}$ to both sides of the equation.

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

Step 4.2.2

Combine the numerators over the common denominator.

$y = \frac{- (2 x \sqrt{6} + \sqrt{6}) + \sqrt{6}}{2}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Apply the distributive property.

$y = \frac{- (2 x \sqrt{6}) - \sqrt{6} + \sqrt{6}}{2}$

Step 4.2.3.2

Multiply $2$ by $- 1$ .

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

Step 4.2.4

Combine the opposite terms in $- 2 x \sqrt{6} - \sqrt{6} + \sqrt{6}$ .

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Step 4.2.4.1

Add $- \sqrt{6}$ and $\sqrt{6}$ .

$y = \frac{- 2 x \sqrt{6} + 0}{2}$

Step 4.2.4.2

Add $- 2 x \sqrt{6}$ and $0$ .

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

Step 4.2.5

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

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Step 4.2.5.1

Factor $2$ out of $- 2 x \sqrt{6}$ .

$y = \frac{2 (- x \sqrt{6})}{2}$

Step 4.2.5.2

Cancel the common factors.

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Step 4.2.5.2.1

Factor $2$ out of $2$ .

$y = \frac{2 (- x \sqrt{6})}{2 (1)}$

Step 4.2.5.2.2

Cancel the common factor.

$y = \frac{2 (- x \sqrt{6})}{2 \cdot 1}$

Step 4.2.5.2.3

Rewrite the expression.

$y = \frac{- x \sqrt{6}}{1}$

Step 4.2.5.2.4

Divide $- x \sqrt{6}$ by $1$ .

$y = - x \sqrt{6}$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - x \sqrt{6}$

Point-slope form:

$y - \frac{\sqrt{6}}{2} = - \sqrt{6} \cdot (x + \frac{1}{2})$

Step 6