Question

slope can be expressed as each of the following except:
\\( m = \frac{rise}{run} \\)
\\( m = \frac{x_2 - x_1}{y_2 - y_1} \\)
\\( m = \frac{vertical\\ change}{horizontal\\ change} \\)
\\( m = \frac{y_2 - y_1}{x_2 - x_1} \\)

Explanation:

Step1: Recall Slope Definitions

Slope \( m \) is defined as \( \frac{\text{rise}}{\text{run}} \), where rise is vertical change (\( y_2 - y_1 \)) and run is horizontal change (\( x_2 - x_1 \)). So \( m=\frac{\text{vertical change}}{\text{horizontal change}}=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Analyze Each Option

• Option 1: \( m = \frac{\text{rise}}{\text{run}} \) is correct (rise = vertical, run = horizontal).
• Option 2: \( m = \frac{x_2 - x_1}{y_2 - y_1} \) swaps horizontal and vertical changes, which is incorrect.
• Option 3: \( m = \frac{\text{vertical change}}{\text{horizontal change}} \) is correct.
• Option 4: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) is correct (standard slope formula).

Answer:

\( m = \frac{x_2 - x_1}{y_2 - y_1} \) (the second option)