Precalculus Examples

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

Step 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 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 3

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Multiply the numerator and denominator of the fraction by $2$ .

Tap for more steps...

Step 4.1.1

Multiply $\frac{- \frac{1}{2} - \frac{1}{2}}{- \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}}$ by $\frac{2}{2}$ .

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

Step 4.1.2

Combine.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify by cancelling.

Tap for more steps...

Step 4.3.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 4.3.1.2

Cancel the common factor.

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

Step 4.3.1.3

Rewrite the expression.

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

Step 4.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.3.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

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

Step 4.3.3.2

Cancel the common factor.

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

Step 4.3.3.3

Rewrite the expression.

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

Step 4.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.4.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

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

Step 4.3.4.2

Cancel the common factor.

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

Step 4.3.4.3

Rewrite the expression.

$m = \frac{- 1 - 1}{- \sqrt{3} - \sqrt{3}}$

Step 4.4

Simplify terms.

Tap for more steps...

Step 4.4.1

Subtract $1$ from $- 1$ .

$m = \frac{- 2}{- \sqrt{3} - \sqrt{3}}$

Step 4.4.2

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$m = \frac{- 2}{- 2 \sqrt{3}}$

Step 4.4.3

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 4.4.3.1

Cancel the common factor.

$m = \frac{- 2}{- 2 \sqrt{3}}$

Step 4.4.3.2

Rewrite the expression.

$m = \frac{1}{\sqrt{3}}$

Step 4.5

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$m = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.6

Combine and simplify the denominator.

Tap for more steps...

Step 4.6.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$m = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$m = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$m = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.6.4

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

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

Step 4.6.5

Add $1$ and $1$ .

$m = \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.6.6

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

Tap for more steps...

Step 4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 4.6.6.2

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

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

Step 4.6.6.3

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

$m = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.6.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.6.6.4.1

Cancel the common factor.

$m = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$m = \frac{\sqrt{3}}{3}$

Step 4.6.6.5

Evaluate the exponent.

$m = \frac{\sqrt{3}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$m = \frac{\sqrt{3}}{3}$

Decimal Form:

$m = 0.57735026 \dots$

Step 6