Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. (4, 7) and (8, 10)

Verified step by step guidance

1. Identify the coordinates of the two points given: Point 1 is \((4, 7)\) and Point 2 is \((8, 10)\).
2. Recall the formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\): \[m = \frac{y_2 - y_1}{x_2 - x_1}\]
3. Substitute the coordinates into the slope formula: \[m = \frac{10 - 7}{8 - 4}\]
4. Simplify the numerator and denominator separately to find the slope fraction: \[m = \frac{3}{4}\]
5. Interpret the slope: since \(m\) is positive and a fraction, the line rises as it moves from left to right.

Key Concepts

Slope of a Line

The slope measures the steepness and direction of a line, calculated as the ratio of the change in y-values to the change in x-values between two points. It is found using the formula m = (y2 - y1) / (x2 - x1). A positive slope indicates the line rises, while a negative slope means it falls.

Undefined Slope

A slope is undefined when the line is vertical, meaning the x-values of both points are the same. Since division by zero is undefined, the slope formula cannot be applied, and the line does not rise or fall but goes straight up and down.

Interpreting Slope Direction

The sign and value of the slope determine the line's direction: positive slope means the line rises from left to right, negative slope means it falls, zero slope indicates a horizontal line, and undefined slope corresponds to a vertical line.