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Chapter 3: Problem 40

Given two points, find the equation of the line. $$ (-5,2),(3,2) $$

Short Answer

Expert verified
The equation of the line is \(y = 2\).

Step by step solution

01

Understanding the problem

We need to find the equation of a line given two points, which are \((-5, 2)\) and \((3, 2)\). A line's equation can be found using the slope-intercept form or point-slope form. Since these points have the same \(y\)-value, it's a special case.
02

Determine the slope

The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points: \(m = \frac{2 - 2}{3 - (-5)} = \frac{0}{8} = 0\). The slope is zero, indicating a horizontal line.
03

Identify the line equation

For a horizontal line with slope zero, the equation is simply \(y = k\), where \(k\) is the constant \(y\)-value of any point on the line. Here, both given points have \(y = 2\), so the equation is \(y = 2\).

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Key Concepts

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

Line Equation
Understanding the concept of a line equation is crucial when dealing with geometry and algebra. A line equation provides a mathematical description of a line in the coordinate plane. There are various forms of line equations, but the two most common are the slope-intercept form and the point-slope form.

  • The **slope-intercept form** is expressed as \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept, the point where the line crosses the y-axis.
  • The **point-slope form** is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Using these equations, you can easily describe and graph a line if you know the slope and at least one point. For lines with specific features, such as a horizontal line, the equation simplifies even further.
Slope Calculation
Calculating the slope of a line is the first step in writing its equation. The slope tells you how steep a line is and in which direction it tilts. Mathematically, the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\\]
  • If the slope \(m\) is positive, the line ascends from left to right.
  • If \(m\) is negative, the line descends from left to right.
  • If \(m = 0\), the line is horizontal.

In the exercise, both y-values are identical, specifically 2. Therefore, the difference in y-values is zero, making the slope zero, and indicating that the line is horizontal.
Horizontal Line
Horizontal lines are a special type of line characterized by their zero slope. This means that the line does not rise or fall as it moves along the x-axis. Instead, it remains constant at a particular y-value.

A horizontal line always takes the form of \(y = k\), where \(k\) represents the constant y-value for every point on the line.

In our specific problem, since both points given are \((鈭�5, 2)\) and \((3, 2)\), the line is not only horizontal but also parallel to the x-axis at \(y = 2\). It's a straightforward example that underlines the concept that no matter how x changes, the y-value remains fixed in a horizontal line.

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