Problem 23 - Understanding Elementary Algebra with Geometry

Question

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

Step-by-Step Solution

Answer

The slope is \(m = -1\).

Step 1: Recall the formula for slope

The slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 2: Substitute the given points into the formula

The given points are \(0, a\) and \(a, 0\). Let \(x_1, y_1\) be \(0, a\), and \(x_2, y_2\) be \(a, 0\). Substitute these into the slope formula: \[ m = \frac{0 - a}{a - 0} \]

Step 3: Simplify the expression

Simplify the expression obtained in Step 2: \[ m = \frac{0 - a}{a - 0} = \frac{-a}{a} = -1 \]

Key Concepts

slope formula

To understand how to find the slope of a line, you first need to know the slope formula. The slope (m) measures the steepness of a line and can be determined using the coordinates of two points on the line. The formula is: (

coordinate geometry

Coordinate geometry helps us understand geometric shapes and their properties using a coordinate system. In this system, points are defined by their coordinates (x, y). When we talk about the slope of a line, we look at how these coordinates change in relation to one another.

simplifying algebraic expressions

Simplifying algebraic expressions is about making complex expressions easier to work with. In the context of finding the slope, this often means reducing fractions or combining like terms. When given points (0, a) and (a, 0) , you substitute these into the slope formula: (