Problem 68 - Algebra 1

Question

MULTIPLE CHOICE The slope of the line passing through the points \((8,0)\) and \((0,8)\) is \(\underline{?}\) \(\begin{array}{llll}\text { (A) positive } & \text { (B) negative } & \text { (C) zero }\end{array}\) (D) undefined

Step-by-Step Solution

Answer

The slope of the line passing through the points (8,0) and (0,8) is negative, or -1.

Step 1: Identify the Points

First, identify the points through which the line passes. The points given are (8,0) and (0,8). Call (8,0) as point 1 (X1,Y1) and (0,8) as point 2 (X2,Y2).

Step 2: Use the Slope Formula

Insert the values from Step 1 into the slope formula: \( m = (Y2 - Y1) / (X2 - X1) \), substituting the coordinates gives: \( m = (8-0) / (0-8) \)

Step 3: Simplify the Result

By simplifying, you obtain: \( m = -8/8 = -1 \). The slope is a negative number, that is -1.

Key Concepts

Linear Equations

A linear equation describes a line on a coordinate plane. It's an equation involving two variables, usually represented by \(x\) and \(y\). An example is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

Here's what each part represents:

• \(m\): Slope, which determines the steepness or angle of the line.
• \(b\): Y-intercept, the point where the line crosses the y-axis.

The slope tells us how much \(y\) changes for each unit of \(x\). In simpler terms, it shows the direction and steepness of the line. A positive \(m\) means the line goes upward as it moves from left to right. Conversely, a negative \(m\) means it goes downward.

Understanding linear equations is crucial for solving problems involving lines, such as determining where two lines intersect or finding the equation of a line passing through two points.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This means we use numbers to define points on a plane. The most common is the Cartesian coordinate system, where each point has an x-value and a y-value, written as \((x, y)\).

The main concept here is that:

• Horizontal lines: have zero slope because there's no vertical change.
• Vertical lines: have an undefined slope as the horizontal change is zero, causing division by zero.
• Other lines can be described with a linear equation using points on the line.

Coordinate geometry helps us make the connection between algebraic formulas and geometric shapes. It allows us to visually represent these shapes on a graph, making it easier to understand their properties and relationships.

With the equation of a line, you can predict how the line behaves and how it interacts with other geometric figures. Whether solving equations or graphing them, coordinate geometry provides powerful tools to unlock these mysteries.

Negative Slope

A negative slope means the line on a graph moves downward as it travels from left to right. This is a key characteristic indicating that the two variables \(x\) and \(y\) have an inverse relationship.

When using the slope formula, \(m = (Y2 - Y1) / (X2 - X1)\), a negative result implies that as one value increases, the other decreases. For instance, in the original exercise with points \((8,0)\) and \((0,8)\), the slope \(m = -1\) tells us that for each unit increase in \(x\), \(y\) decreases by the same amount.

Negative slopes are commonly encountered in various real-world scenarios, such as declining sales over time or temperatures dropping with decreasing daylight. Understanding negative slopes helps interpret such trends effectively.

To graph a negative slope, simply start from a point and move downwards while moving to the right, following the slope's ratio. This visual representation helps to intuitively grasp how the negative slope reflects the inverse relationship between \(x\) and \(y\).