Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-40: Find an equation for the line that bisects the given quadrants. $$ \text { I and III } $$

Step-by-Step Solution

Answer

The equation for the line is \(y = x\).

Step 1: Understanding the problem

To find the line that bisects quadrants I and III, we need to recognize that such a line will pass through the origin \((0,0)\) and divide the plane into equal areas in these quadrants. Quadrant I includes points where both \(x\) and \(y\) are positive, and quadrant III includes points where both \(x\) and \(y\) are negative.

Step 2: Visualizing the line

The line that perfectly bisects quadrants I and III is a diagonal line that makes a 45-degree angle with the positive x-axis. This is because each point in quadrant I has its corresponding negative point in quadrant III, reflecting across the x and y axes.

Step 3: Equation of the line

The line at a 45-degree angle passing through the origin has the equation \(y = x\), as the slope \(m\) of this line is 1 (rise over run equals 1 over 1). This means for every unit increase in \(x\), \(y\) increases by the same amount.

Key Concepts

Lines and Slopes

In analytic geometry, understanding lines and slopes is crucial to learning how geometric figures are represented algebraically. A line is simply a straight path extending indefinitely in both directions, without thickness. What makes each line unique in math is its slope, denoted by 'm'. The slope of a line measures how steep the line is. It's calculated as the ratio of the vertical change (known as rise) to the horizontal change (called run) between two points on the line. Mathematically, this is expressed as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Where \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line. A positive slope means the line is ascending from left to right, whereas a negative slope indicates a descending line. Slopes of zero are horizontal lines, and undefined slopes are vertical lines. Understanding slopes helps us appreciate various line orientations on the coordinate plane.

Quadrants in Coordinate Plane

The coordinate plane is divided into four regions called quadrants, which help locate points in two-dimensional space. These quadrants are determined by the x-axis (horizontal) and y-axis (vertical). Each quadrant has distinct conditions based on the signs of the coordinates (x, y):

• Quadrant I: Both x and y are positive (top right).
• Quadrant II: x is negative, y is positive (top left).
• Quadrant III: Both x and y are negative (bottom left).
• Quadrant IV: x is positive, y is negative (bottom right).

These quadrants are numbered in an anti-clockwise direction. Each point on the coordinate plane can be identified by its quadrant, which gives insight into the point's position and relationship to the axes.

Equation of a Line

The equation of a line is a fundamental concept in analytic geometry that represents a straight path on the coordinate plane. The most common form is the slope-intercept form, expressed as:\[y = mx + b\]Here, \(m\) denotes the slope of the line, representing how much y changes for a step in x, and \(b\) is the y-intercept, the point where the line crosses the y-axis. When a line passes through the origin, like the one bisecting Quadrants I and III, the y-intercept \(b\) is zero, simplifying the equation to:\[y = x\]This specific line has a slope of 1, or a 45-degree angle with the positive x-axis, making it equidistant from both x and y axes. Understanding these equations is key to graphing lines and analyzing their interactions within the plane.