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Chapter 1: Problem 5

Find the slopes and \(y\)-intercepts of the following lines. $$y=\frac{x}{7}-5$$

Short Answer

Expert verified
Slope = \( \frac{1}{7} \), Y-Intercept = \( -5 \)

Step by step solution

01

Identify the Slope-Intercept Form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where \( m \) is the slope, and \( b \) is the y-intercept.
02

Extract the Slope

Compare the given equation \( y = \frac{x}{7} - 5 \) with the slope-intercept form \( y = mx + b \). Here, \( m \) (the coefficient of \( x \)) represents the slope. Therefore, the slope \( m \) is: \[ m = \frac{1}{7} \]
03

Find the Y-Intercept

In the equation \( y = \frac{x}{7} - 5 \), the constant term \( -5 \) represents the y-intercept. So, the y-intercept \( b \) is: \[ b = -5 \]
04

Final Answer

The slope of the line is \( \frac{1}{7} \), and the y-intercept is \( -5 \).

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Key Concepts

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

slope-intercept form
To understand the equation of a line, we first need to get familiar with the slope-intercept form. This form is written as: \[ y = mx + b \]In this equation, \(m\) stands for the slope of the line, which tells us how steep the line is. The \(b\) is the y-intercept, showing where the line crosses the y-axis. This format is very handy as it quickly tells us the key features of the line.
For example, in the equation \(y = \frac{x}{7} - 5\), we can see it is already in the slope-intercept form. Let's break down the parts so you can see how they fit together. It makes our job of identifying the slope and y-intercept much easier, just by comparing it to \(y = mx + b\).
slope
The slope of a line is all about how the line inclines or declines as it moves from left to right. It is found by the coefficient of \(x\) in our equation.
In the given equation \( y = \frac{x}{7} - 5 \), we identify the slope by comparing it with the slope-intercept form \( y = mx + b \). Here, the term \(\frac{x}{7}\) can be rewritten as \( \frac{1}{7}x\), indicating that the slope \(m\) is \( \frac{1}{7} \).
So what does \( \frac{1}{7} \) tell us?
  • If the slope is a positive number like \( \frac{1}{7} \), the line rises as it moves to the right.
  • If the slope were negative, the line would fall as you move to the right.
The fraction \( \frac{1}{7} \) means the line is not very steep.
y-intercept
The y-intercept is the point where the line crosses the y-axis. This happens when \(x\) is 0. In the slope-intercept form \( y = mx + b \), the y-intercept is the \(b\) term.
For our equation \( y = \frac{x}{7} - 5 \), you can see that the y-intercept \(b\) is \(-5\). This tells us that when \(x\) is 0, \(y\) is \(-5\).
Visually, this means the line crosses the y-axis at the point (0, -5).
  • If the y-intercept is positive (like \(+3\)), it means the line crosses the y-axis above the origin (0, 0).
  • If it鈥檚 negative (like \(-5\) here), the crossing point is below the origin.
The y-intercept gives us a clear starting point for drawing the line on the graph.

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Most popular questions from this chapter

Compute the difference quotient $$\frac{f(x+h)-f(x)}{h}.$$ Simplify your answer as much as possible. $$f(x)=2 x^{3}+x^{2}$$

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. $$f^{\prime}(1), \text { where } f(x)=\frac{1}{1+x^{2}}$$

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=\sqrt{x^{2}+1}$$

A toy company introduces a new video game on the market. Let \(S(x)\) denote the number of videos sold on the day, \(x,\) since the item was introduced. Let \(n\) be a positive integer. Interpret \(S(n), S^{\prime}(n),\) and \(S(n)+S^{\prime}(n)\)

Let \(y\) denote the percentage of the world population that is urban \(x\) years after 2014. According to data from the United Nations, 54 percent of the world's population was urban in 2014 , and projections show that this percentage will increase to 66 percent by 2050. Assume that \(y\) is a linear function of \(x\) since 2014. (a) Determine \(y\) as a function of \(x.\) (b) Interpret the slope as a rate of change. (c) Find the percentage of the world's population that is urban in 2020. (d) Determine the year in which \(72 \%\) of the world's population will be urban.
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