Question

True or false? Give an explanation for your answer. If $y$ is a linear function of $x,$ then the ratio $y / x$ is constant for all points on the graph at which $x \neq 0$.

Name: Problem 83, Chapter 1: Calculus Single Variable
Uploaded: 2021-05-04T12:49:57-08:00
Duration: 1 min 42 s
Channel: Gregory Higby
Description: True or false? Give an explanation for your answer. If y is a linear function of x, then the ratio y / x is constant for all points on the graph at which x ≠0.

   True or false? Give an explanation for your answer.
If $y$ is a linear function of $x,$ then the ratio $y / x$ is constant for all points on the graph at which $x \neq 0$.

Calculus Single Variable

Deborah… 7th Edition

Chapter 1, Problem 83

Instant Answer

Solved by Expert Gregory Higby

05/04/2021

Step 1

This means that $y$ can be expressed in the form of a linear equation, which is $y = mx + b$. Show more…

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True or false? Give an explanation for your answer. If $y$ is a linear function of $x,$ then the ratio $y / x$ is constant for all points on the graph at which $x \neq 0$.

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Key Concepts

Linear Function

A linear function is a function of the form y = mx + b, where m is the slope and b is the y-intercept. This represents a straight-line graph in the Cartesian plane and indicates a constant rate of change of y with respect to x.

Constant Ratio y/x

The ratio y/x, when expressed for a linear function y = mx + b, becomes m + (b/x). This ratio is only constant if the y-intercept b is zero; otherwise, it varies with x. This distinguishes functions that are directly proportional (y = kx) from general linear functions.

Direct Proportionality

Direct proportionality is a special case of linear functions where the y-intercept is zero, resulting in y = kx. In this case, the ratio y/x is always equal to the constant k, which is not true for general linear functions with a nonzero y-intercept.

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