Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. (-a, 0) and (0, -b)

Verified step by step guidance

1. Identify the coordinates of the two points given: the first point is \((-a, 0)\) and the second point is \((0, -b)\).
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: \(m = \frac{-b - 0}{0 - (-a)}\).
4. Simplify the numerator and denominator: \(m = \frac{-b}{a}\).
5. Analyze the slope value: since \(a\) and \(b\) are positive, the slope \(m\) is negative, which means the line falls as it moves from left to right.

Key Concepts

Slope of a Line

The slope of a line measures its steepness and is calculated as the ratio of the change in y-values to the change in x-values between two points. It is given by the formula m = (y2 - y1) / (x2 - x1). This value indicates how much y changes for a unit change in x.

Undefined Slope and Vertical Lines

A slope is undefined when the denominator in the slope formula (change in x) is zero, meaning the line is vertical. Vertical lines have the form x = constant and do not have a defined slope because division by zero is undefined.

Interpreting Slope Sign: Rising, Falling, Horizontal, Vertical

The sign of the slope indicates the line's direction: positive slope means the line rises from left to right, negative slope means it falls, zero slope means the line is horizontal, and undefined slope means the line is vertical. This helps describe the line's orientation visually.