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Chapter 3: Problem 10

Determine whether each equation is linear. $$ 3 y=4 x $$

Short Answer

Expert verified
Yes, the equation \(3y = 4x\) is linear.

Step by step solution

01

Understand the Definition of a Linear Equation

A linear equation is an equation that forms a straight line when plotted on a graph. It can be written in the form \[y = mx + b\] where \(m\) and \(b\) are constants.
02

Rewrite the Given Equation

Given the equation is \(3y = 4x\). To determine if it is linear, rewrite it in the form of \(y = mx + b\).
03

Solve for y

Divide both sides of the equation by 3 to solve for \(y\): \[\frac{3y}{3} = \frac{4x}{3}\] Simplifies to \[y = \frac{4}{3}x\]
04

Compare to the Linear Equation Form

The rewritten equation is \(y = \frac{4}{3}x\), which matches the form \(y = mx + b\) with \(m = \frac{4}{3}\) and \(b = 0\).
05

Conclusion

Since the equation \(y = \frac{4}{3}x\) matches the form of a linear equation, it is indeed a linear equation.

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Key Concepts

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

Definition of Linear Equation
A linear equation is fundamental in algebra and represents a straight line graphically. It is often written in the form \( y = mx + b \) where:
  • \( y \) is the dependent variable.
  • \( x \) is the independent variable.
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
This form ensures the equation models a straight line. For an equation to be linear, it should not have variables multiplied by each other or raised to any exponent other than one. Hence, terms like \( x^2 \) or \( xy \) are not allowed.
Solving for y
To determine if an equation is linear, one often needs to solve for \( y \) to compare it to the standard linear form \( y = mx + b \). Let's take a step-by-step approach using the given example:
Given: \( 3y = 4x \). 1. Divide both sides by 3 to isolate \( y \): \( \frac{3y}{3} = \frac{4x}{3} \) This simplifies to: \( y = \frac{4}{3} x \). By solving for \( y \), we now have the equation in a recognizable linear form where the slope \( m \) is \( \frac{4}{3} \) and the y-intercept \( b \) is 0. This process shows how isolating \( y \) helps in identifying the characteristics of the equation, making it easier to determine if it鈥檚 linear.
Rewriting Equations
Rewriting equations is a valuable skill in algebra. It allows us to see different representations of an equation and to manipulate it into the desired form. For instance, given the equation: \( 3y = 4x \)
  • First, identify what needs to be isolated. Here, we want \( y \) on one side
  • To isolate \( y \), divide both sides by 3 as shown earlier: \( \frac{3y}{3} = \frac{4x}{3} \)
The result is \( y = \frac{4}{3} x \), clearly in the form \( y = mx + b \). By rewriting, you make it simpler to recognize the type and properties of the equation. Here we proved the equation to be linear. Rewriting is thus a powerful technique not just for solving equations but also for understanding their nature.

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