/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Determine whether each equation ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 14

Determine whether each equation is linear. $$ 3 x-12=0 $$

Short Answer

Expert verified
Yes, the equation is linear.

Step by step solution

01

Understand the Definition of a Linear Equation

A linear equation is an equation that can be written in the form \[ ax + b = 0 \] where \( a \) and \( b \) are constants, and \( x \) is the variable. It represents a straight line graphically.
02

Identify Constants and Variables

In the given equation, \( 3x - 12 = 0 \), identify \( a \) and \( b \). Here, \( a = 3 \) and \( b = -12 \), and \( x \) is the variable.
03

Compare with the Standard Linear Form

Compare the given equation \( 3x - 12 = 0 \) with the standard linear form \( ax + b = 0 \). Verify whether it matches the criteria.
04

Conclusion

Since the equation \( 3x - 12 = 0 \) can be written in the form \( ax + b = 0 \), it is indeed a linear equation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Constants and Variables
In any equation, you will come across constants and variables. Understanding these two concepts is crucial for solving mathematical problems.
Constants are fixed values that do not change, no matter the circumstances. For example, in the equation \(3x - 12 = 0\), the numbers 3 and -12 are constants.
On the contrary, variables can change. In the same equation, the letter \(x\) represents an unknown value that we want to find.
To distinguish between constants and variables:
  • Constants: Fixed Numbers
  • Variables: Unknowns that can take different values
Learning to quickly identify constants and variables helps in comparing equations and solving them efficiently.
Standard Linear Form
A linear equation can be identified by its standard form. The standard linear form is written as \[ax + b = 0\], where:
  • \(a\) is the coefficient of the variable \(x\)
  • \(b\) is the constant term
  • \(x\) is the variable
For example, in the equation \(3x - 12 = 0\), we can see:
  • \(a = 3\)
  • \(b = -12\)
  • \(x\) is the variable
To check if an equation is linear, see if it can be rearranged into the form \[ax + b = 0\]. If it can, then it's a linear equation. This form not only makes the equation easier to work with but also confirms its linearity.
Graph of a Linear Equation
Graphing a linear equation helps visualize its behavior. A linear equation like \[3x - 12 = 0\] represents a straight line when plotted on a Cartesian plane.
To plot this equation, find the points where the line crosses the axes. For example:
  • When \(x = 0\), solve for \(y\).
  • When \(y = 0\), solve for \(x\).
In our example equation \(3x - 12 = 0\), setting \(x = 0\) gives \(-12 = 0\), which doesn't provide a valid point. Therefore, we solve \(3x - 12 = 0\) for \(x\):
\[x = \frac{12}{3} = 4\]
This means the line crosses the \(x\)-axis at (4, 0).
This method quickly helps students understand the linear nature and visualize the equation as a straight line.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Marcy picks apples twice as fast as Whitney does. By \(4: 30,\) Whitney had already picked 4 bushels of apples. Fifty minutes later, her total reached \(5 \frac{1}{2}\) bushels. Find Marcy's picking rate. Give your answer in number of bushels per hour.

Running. Jerod ran from the 2 -mi marker to the finish line of a 5 -mi race in 25 min. At this rate, how long would it take Jerod to run a \(10-\mathrm{km}\) race? (Hint: \(\mathrm{mi} \approx 1.61 \mathrm{km} .\)

Assume that \(r, p,\) and \(s\) are constants and that \(x\) and \(y\) are variables. Determine the slope and the y-intercept. $$r x+p y=s$$

If \(f(x)=0.1 x-0.5,\) for what input is the output \(-3 ?\)

To prepare for Section \(3.8,\) review evaluating algebraic expressions (Section \(1.8)\) $$ 2-x, \text { for } x=-3 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.