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Chapter 12: Problem 60

Find the equation of the line through the given points. $$(2,-1) \text { and }(5,-1)$$

Short Answer

Expert verified
The equation of the line passing through the points (2,-1) and (5,-1) is \(y = -1\)

Step by step solution

01

Calculating the Slope

The slope of a line through the points \((x_{1}, y_{1})\) and \((x_{2},y_{2})\) is calculated as \(m = (y_{2} - y_{1}) / (x_{2} - x_{1})\). Substituting the given points \((2,-1)\) and \((5,-1)\) into the formula, we find that \(m = (-1 - (-1)) / (5 - 2) = 0\) which is the slope of a horizontal line.
02

Equation of the Line

The equation of a line is typically given in the format y=mx+c. However, given the slope of this line is 0, the y-value remains constant. This means that the equation of the line can be written as y = c. Considering that the points lie on the line, we can substitute one of the y-values for c. For the given points \((2,-1)\) and \((5,-1)\), the y-values are each -1. Hence, the equation of the line is \(y =-1\).

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Key Concepts

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

Slope
The slope of a line indicates its steepness and direction. When we talk about the slope, it refers to the ratio of the vertical change, or "rise," to the horizontal change, or "run," between two points on the line.
  • If the slope is positive, the line ascends from left to right.
  • A negative slope means the line descends from left to right.
  • A slope of zero indicates a horizontal line since there is no rise, only run.
In the exercise, the slope is calculated using the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Given the points \((2, -1)\) and \((5, -1)\), the slope is \( m = \frac{(-1) - (-1)}{5 - 2} = 0 \). This confirms that the line is horizontal.
Horizontal Line
A horizontal line is a straight line that extends left to right in the same direction as the x-axis. One of the main characteristics of a horizontal line is that it has a slope of zero. There is no vertical change when moving along this line.In other words, regardless of the change in the x-coordinate, the y-coordinate remains constant. When you see an equation such as \( y = -1 \), it represents a horizontal line meaning every point on this line will have a y-value of \(-1\). Therefore, all points will be like
  • (x, -1)
Understanding this concept ensures that you can easily identify horizontal lines in any linear equation problem.
Coordinates
Coordinates are used to describe the specific position of a point on a plane. Each point in a two-dimensional space is represented by a pair of numbers known as an ordered pair \((x, y)\). Each number in this pair signifies a specific position on the x-axis and the y-axis respectively.
  • The first number is the x-coordinate, which measures horizontal movement from the origin.
  • The second number is the y-coordinate, which measures vertical movement from the origin.
In the given exercise, the coordinates \((2, -1)\) and \((5, -1)\) tell us where the points are located. These points lie on the line described by the equation \(y = -1\). Understanding how to read and use coordinates is foundational in working with graphs and equations.
Linear Equations
Linear equations describe a straight line on a graph and can be written in the form \(y = mx + c\), where \(m\) is the slope, and \(c\) is the y-intercept. This equation illustrates the relationship between x and y in a linear, proportional fashion.
  • The y-intercept, \(c\), is where the line crosses the y-axis.
  • In the case of horizontal lines, the equation simplifies to \(y = c\) since the slope \(m = 0\).
For our example, the equation \(y = -1\) represents a horizontal line, indicating that regardless of the value of \(x\), \(y\) will always equal \(-1\). Understanding linear equations helps in determining how variables relate to each other, especially in graphing contexts.

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