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Chapter 3: Problem 194

In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form. $$ (-6,-3) \text { and }(-1,-3) $$

Short Answer

Expert verified
The equation of the line is \( y = -3 \).

Step by step solution

01

- Find the slope

Use the formula for the slope between two points, which is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the given points \[ (-6, -3) \text{ and } (-1, -3) \] into the formula: \[ m = \frac{-3 - (-3)}{-1 - (-6)} = \frac{0}{5} = 0 \] The slope is 0.
02

- Understand the slope result

A slope of 0 means the line is horizontal. For any horizontal line, the y-coordinate remains constant.
03

- Find the y-coordinate

Since both given points have a y-coordinate of -3, the equation of the line is simply \( y = -3 \).

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Key Concepts

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

slope-intercept form
When dealing with lines in algebra, it's very useful to know the slope-intercept form of a line. This form is written as: \[ y = mx + b \]
  • \(y\) represents the y-coordinate of any point on the line.
  • \(x\) represents the x-coordinate of any point on the line.
  • \(m\) is the slope of the line, which shows how steep the line is.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
The slope-intercept form makes it easy to graph the line and understand its behavior. You can quickly see where the line will cross the y-axis and how it will move. It's a must-understand concept for any student studying lines and their equations.
horizontal line
Horizontal lines are special types of lines in mathematics. These lines have a unique property: they are completely flat and do not rise or fall. Therefore, their slope is zero. For any line that is horizontal, the y-coordinate remains constant, meaning all points on the line will share the same y-value. This can be easily recognized in the equation of a horizontal line which is: \[ y = c \] Here, \(c\) is the constant y-value that all points on the line share. In this exercise, our line passed through points \((-6, -3)\) and \((-1, -3)\). Both points share the same y-coordinate of \(-3\), indicating it's a horizontal line.
slope formula
The slope of a line is a measure of its steepness and direction. To find the slope of a line that passes through two points, you can use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] \(m\) represents the slope, while
  • \(x_1,
  • y_1\) are the coordinates of the first point,
  • \(x_2,
  • y_2\) are the coordinates of the second point.
This slope formula tells you how much the y-coordinate changes for a change in the x-coordinate. For our given points: \((-6, -3)\) and \((-1, -3)\), substituting them into the slope formula gives: \[ m = \frac{-3 - (-3)}{-1 - (-6)} = \frac{0}{5} = 0 \] This result shows the slope is zero, confirming that our line is horizontal. Understanding how to use the slope formula is essential for determining the nature of a line connecting two points.

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