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Chapter 3: Problem 69

Find the equation of the line given two points on the line. (7,-6) and (3,-6)

Short Answer

Expert verified
The equation of the line is \(y = -6\).

Step by step solution

01

Identify the coordinates of the given points

The given points are (7,-6) and (3,-6). The coordinates of the first point are \((x_1, y_1) = (7, -6)\) and the coordinates of the second point are \((x_2, y_2) = (3, -6)\).
02

Calculate the slope (m) of the line

The formula for the slope \(m\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the coordinates of the points: \( m = \frac{-6 - (-6)}{3 - 7} = \frac{0}{-4} = 0 \). Since the slope is zero, the line is horizontal.
03

Determine the equation of the line

For a horizontal line, the equation is simply \(y = b\) where \(b\) is the constant y-value for any point on the line. Since both given points have a y-coordinate of -6, the equation is \(y = -6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope of a Line
The slope of a line is a measurement of its steepness and direction. To calculate the slope, also represented by the letter \(m\), you use the formula:
  • \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
In this formula, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line.
This equation calculates how much the y-coordinate changes for every change in the x-coordinate. If a line goes upwards as you move right, the slope is positive. Conversely, if it goes downwards, the slope is negative.
A zero slope indicates the line is completely flat, like in this exercise where both given points have the same y-value of -6.
This flatness results in a slope of zero, confirming the line is perfectly horizontal.
Horizontal Line
A horizontal line is unique because it has no vertical change between any two points on the line. This means the slope is always zero. The general equation for a horizontal line is simple:
  • \(y = b\)
Here, \(b\) represents the constant y-value for any point on the line.
In other words, no matter what x-value you choose, the y-value will always remain fixed. For the points (7, -6) and (3, -6), since each point shares the same y-coordinate, -6, this constant value is used in the equation, making it \(y = -6\).
This consistency implies that every point on this line has a y-coordinate of -6, producing a perfectly straight, horizontal line.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves using a coordinate system to solve geometric problems. It provides a bridge between algebra and geometry through graphs and Cartesian coordinates.In this branch of mathematics:
  • Points are defined using coordinates, such as \((x, y)\).
  • Lines can be represented with equations, like \(y = mx + b\).
  • The distance between points, midpoints, and line gradients are calculated.
Coordinate geometry helps us understand the spatial relationships between points and lines.
This allows us to find missing coordinates, determine distances, and solve other geometric problems.
For instance, in this exercise, understanding how points relate on a coordinate plane lets us derive the equation for the line containing those points, enriching our comprehension of geometric concepts using algebraic methodologies.

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