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Chapter 1: Problem 111

Explain why \(\frac{3}{x}+12=0\) is not a linear equation in one variable.

Short Answer

Expert verified
The term \( \frac{3}{x} \) is \( 3x^{-1} \), which has \( x \) raised to the power \( -1 \). Linear equations must have the variable raised to the first power only.

Step by step solution

01

Understand the Definition of a Linear Equation

A linear equation in one variable has the general form \( ax + b = 0 \), where \( a \) and \( b \) are constants and \( x \) is the variable. The equation must have the variable \( x \) raised to the first power only.
02

Analyze the Given Equation

Consider the given equation \( \frac{3}{x} + 12 = 0 \). The term \( \frac{3}{x} \) can be rewritten as \( 3x^{-1} \).
03

Identify the Variable's Exponent

Notice that in the term \( 3x^{-1} \), the variable \( x \) is raised to the power of \( -1 \), which is not the first power. This implies that the equation is not in the form of \( ax + b = 0 \).
04

Conclusion

Since the equation \( \frac{3}{x} + 12 = 0 \) includes a term with \( x \) raised to the power of \( -1 \), it is not a linear equation in one variable.

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Key Concepts

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

linear equation definition
To understand why the given equation is not a linear equation in one variable, we must first define what a linear equation is. A linear equation in one variable is an algebraic equation that can be written in the form:
\[ ax + b = 0 \]
Here, a and b are constants, and x is the variable. The key characteristic is that the variable x must be raised to the first power only. This means you should not see exponents other than one attached to the variable. For example, equations like 2x + 3 = 0 or -4x + 7 = 0 are all linear equations because they fit the described form.
variable exponent
Now, let us analyze the given equation:
\[ \frac{3}{x} + 12 = 0 \]
The term \(\frac{3}{x}\) can be rewritten as 3x^{-1} to better understand its form. Here, rather than being raised to the first power, the variable x is raised to the power of -1. This alteration means the equation no longer meets the criteria for a linear equation. Remember, a linear equation requires that the variable's exponent is exactly one. Any deviation, whether higher or lower, disqualifies it from being a linear equation.
one-variable equation analysis
Finally, let's delve deeper into why the equation \(\frac{3}{x} + 12 = 0\) is not a linear equation in one variable. A linear equation in one variable involves simplifying the equation to the form ax + b = 0, where x is in the first degree. In this case, 3x^{-1} indicates an inverse relationship. Such terms introduce complexity that prevents the equation from being linear. To identify non-linear equations, check if the variable is raised to any power other than one.
Therefore, because the variable exponent in \(\frac{3}{x}\) is -1, the equation \(\frac{3}{x} + 12 = 0\) does not qualify as a linear equation in one variable. Understanding this distinction helps categorize and solve different types of equations effectively.

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