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Chapter 0: Problem 45

Write the equation of the line, given the slope and intercept. Slope: \(m=-\frac{1}{3}\) \(y\) -intercept: (0,0)

Short Answer

Expert verified
The equation of the line is \(y = -\frac{1}{3}x\).

Step by step solution

01

Understanding the Formula

To write the equation of a line in slope-intercept form, we use the formula: \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept.
02

Substitute the Given Values

In this problem, we know the slope \(m\) is \(-\frac{1}{3}\) and the \(y\)-intercept is the point (0,0), which means \(b=0\). Substitute these values into the equation: \(y = -\frac{1}{3}x + 0\).
03

Simplify the Equation

Since adding zero does not change the equation, it simplifies to \(y = -\frac{1}{3}x\). This is a straight line passing through the origin with a negative slope.

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Key Concepts

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

Slope-Intercept Form
One of the most common ways to express the equation of a line is through the slope-intercept form. This formula is particularly user-friendly as it quickly reveals the key characteristics of a line 鈥� the slope and the y-intercept 鈥� at a glance. The equation has the structure:
  • \(y = mx + b\)
The variable \(m\) represents the slope of the line, while \(b\) stands for the y-intercept. The simplicity of this form is that it allows you to identify the steepness and direction of the line and its specific point of crossing the y-axis directly from the equation. When a problem provides the slope and y-intercept, this form ensures you can easily construct the equation of the line.
Slope of a Line
The slope of a line is a measure of its steepness and direction. Think of it as the rate of change. In mathematical terms, the slope is the ratio of the 'rise' (vertical change) over the 'run' (horizontal change) between any two points on the line.
  • A positive slope means the line inclines upward as it moves from left to right.
  • A negative slope means it descends.
  • A slope of zero indicates a horizontal line.
  • An undefined slope describes a vertical line.
For this exercise, the slope is given as \(m = -\frac{1}{3}\), indicating that for every 3 units the line moves horizontally, it descends by 1 unit vertically. This negative value clearly suggests that the line slopes downward from left to right.
y-Intercept
The y-intercept of a line is the point where the line intersects the y-axis. It is an essential characteristic because it represents the output value (\(y\)) when the input (\(x\)) is zero. In the slope-intercept form \(y = mx + b\), the \(b\) corresponds to the y-intercept.In this specific problem, the y-intercept is given as \((0,0)\). This indicates that the line not only passes through the origin, but also that the y-intercept value \(b\) is 0. Essentially, this line's equation simplifies broadly, focusing solely on its slope. As a result, one can calculate any point on this line simply through the multiplication of the slope and a given \(x\)-value, enhancing comprehension of line behavior in a graph.

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Most popular questions from this chapter

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