Problem 24 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((0,-b)\) and \((-b, 0), b \neq 0\)

Step-by-Step Solution

Answer

The slope is \(-1\).

Step 1 - Understand the Slope Formula

Recall that the slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\) is given by \[\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\].

Step 2 - Identify Coordinates

For the given points \(0,-b\) and \(-b, 0\), identify \(x_1 = 0, y_1 = -b\) and \(x_2 = -b, y_2 = 0\).

Step 3 - Substitute into the Slope Formula

Substitute the coordinates into the slope formula: \[\text{slope} = \frac{0 - (-b)}{-b - 0} = \frac{b}{-b} = -1\].

Key Concepts

slope formula

The slope of a line tells us how steep the line is. It is determined by the ratio of the vertical change to the horizontal change between two points on the line. The slope formula is easy to use if you follow the steps carefully.
The formula for the slope (m) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula calculates the difference in the y-coordinates divided by the difference in the x-coordinates. This ratio tells us how much y changes for a given change in x.
When you use this formula, always subtract the coordinates in the same order, starting with the second point's coordinates minus the first point's coordinates. The result shows the rate at which the line rises or falls.

coordinate geometry

Coordinate geometry helps us understand the position of points and the relationship between them. In this exercise, we are working with two points: \( (0, -b) \) and \( (-b, 0) \).
Each point has an x-coordinate and a y-coordinate. For \( (0, -b) \), the x-coordinate is 0 and the y-coordinate is \(-b \), which means the point is b units below the origin on the y-axis.
For \( (-b, 0) \), the x-coordinate is \(-b \) and the y-coordinate is 0, meaning the point is b units to the left of the origin on the x-axis.
The slope helps us understand how these points are related to each other on the graph. By substituting these coordinates into the slope formula, we can determine how steep the line is that passes through them.

algebra

Algebra is crucial for finding the slope of a line. It involves manipulating symbols and equations to find the desired value. Let's see how algebra helps in this problem.
First, identify the coordinates of the points: \( (0, -b) \) and \( (-b, 0) \), and substitute these into the slope formula: \[ m = \frac{0 - (-b)}{-b - 0} \] Simplify the numerator and the denominator: \[ m = \frac{b}{-b} \] By dividing b by \(-b \), we get \(-1 \). Thus, the slope of the line is \(-1 \).
Understanding algebra helps us follow these steps and confirm our result. By simplifying expressions and solving equations, we can accurately find the slope.