Algebra Examples

$(1, 7)$ , $(10, 1)$

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.

$m = \frac{1 - (7)}{10 - (1)}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $7$ .

$m = \frac{1 - 7}{10 - (1)}$

Step 4.1.2

Subtract $7$ from $1$ .

$m = \frac{- 6}{10 - (1)}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $- 1$ by $1$ .

$m = \frac{- 6}{10 - 1}$

Step 4.2.2

Subtract $1$ from $10$ .

$m = \frac{- 6}{9}$

Step 4.3

Reduce the expression by cancelling the common factors.

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Step 4.3.1

Cancel the common factor of $- 6$ and $9$ .

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Step 4.3.1.1

Factor $3$ out of $- 6$ .

$m = \frac{3 (- 2)}{9}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $3$ out of $9$ .

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

Step 4.3.1.2.2

Cancel the common factor.

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

Step 4.3.1.2.3

Rewrite the expression.

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

Step 4.3.2

Move the negative in front of the fraction.

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

Step 5

The slope of a perpendicular line is the negative reciprocal of the slope of the line that passes through the two given points.

$m_{perpendicular} = - \frac{1}{m}$

Step 6

Simplify $- \frac{1}{- \frac{2}{3}}$ .

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Step 6.1

Cancel the common factor of $1$ and $- 1$ .

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Step 6.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{2}{3}}$

Step 6.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{2}{3}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{3}{2})$

Step 6.3

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

$m_{perpendicular} = \frac{3}{2}$

Step 6.4

Multiply $- - \frac{3}{2}$ .

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{3}{2})$

Step 6.4.2

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

$m_{perpendicular} = \frac{3}{2}$

Step 7