Problem 38 If \(k \neq 0,\) is the equation... [FREE SOLUTION] (2024)

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Chapter 1: Problem 38

If \(k \neq 0,\) is the equation \(x+k=x\) a contradiction, a conditionalequation, or an identity? Explain.

Short Answer

Expert verified

The equation \( x + k = x \) is a contradiction because it results in \( k = 0 \) while \( k eq 0 \).

Step by step solution

01

- Understanding the Equation

The given equation is: \[ x + k = x \]where \( k eq 0 \).

02

- Isolate the Variable

Subtract \( x \) from both sides of the equation to isolate the variable:\[ x + k - x = x - x \]This simplifies to:\[ k = 0 \].

03

- Analyze the Result

Since \( k eq 0 \), the equation \( k = 0 \) cannot be true.

04

- Determine the Type of Equation

An equation that cannot be true is known as a contradiction. Therefore, the given equation is a contradiction.

Key Concepts

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

Contradiction Equation

An equation is considered a contradiction when it has no solution. This type of equation yields a statement that is always false, regardless of the value of the variables.
In the provided exercise, the equation given is: \( x + k = x \), where \( k eq 0 \).
When we isolate the variable by subtracting \( x \) from both sides, we get: \[ x + k - x = x - x \].
This simplifies to: \[ k = 0 \].
Given that \( k eq 0 \), the statement \( k = 0 \) is impossible. Thus, we cannot find any value of \( x \) that will satisfy the equation.
This leads us to conclude that the equation is a contradiction since it fails to hold true under any circ*mstance. Contradiction equations are useful in identifying when certain assumptions or conditions are inherently conflicting.

Isolating Variables

Isolating variables is a fundamental step in solving most equations. The goal is to get the variable of interest alone on one side of the equation, which allows us to solve for its value.
In the context of the provided exercise, isolating the variable was a key step. The original equation given was: \( x + k = x \), where \( k eq 0 \).
To isolate \( k \), we subtracted \( x \) from both sides of the equation, resulting in: \[ x + k - x = x - x \].
This simplified to: \[ k = 0 \].
Isolating variables can involve various techniques, such as:

  • Addition or subtraction
  • Multiplication or division
  • Combining like terms

It is important to perform the same operation on both sides of the equation to maintain its balance. Once the variable is isolated, we can determine its value or analyze the resulting statement.

Conditional Equations

A conditional equation is an equation that is true for some values of the variable(s) but not for all. The solution set for a conditional equation includes all the values that make the equation true.
For example, the equation \( 2x = 4 \) is conditional because it is only true when \( x = 2 \). For values other than 2, the equation does not hold.
In contrast, the equation in the provided exercise, \( x + k = x \), when \( k eq 0 \), is not conditional but a contradiction.
When isolating the variable \( k \), we found that the resulting statement \( k = 0 \) could never be true under the condition \( k eq 0 \).
Conditional equations are quite common in algebra. Solving them involves

  • Isolating the variable
  • Checking the solution(s) by substituting back into the original equation
  • Verifying that the solutions meet any given constraints or conditions

Understanding the difference between conditional equations, contradictions, and identities is crucial for mastering algebra and precalculus.

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Problem 38 If \(k \neq 0,\) is the equation... [FREE SOLUTION] (3)

Most popular questions from this chapter

If p units of an item are sold for \(x\) dollars per unit, the revenue is \(R=px\). Use this idea to analyze the following problem. Number of Apartments Rented The manager of an 80-unit apartment complex knowsfrom experience that at a rent of \(\$ 300,\) all the units will be full. On theaverage, one additional unit will remain vacant for each \(\$ 20\) increase inrent over \(\$ 300 .\) Furthermore, the manager must keep at least 30 unitsrented due to other financial considerations. Currently, the revenue from the complex is \(\$ 35,000 .\) How many apartmentsare rented?Suppose that \(x\) represents the number of \(\$ 20\) increases over \(\$ 300 .\)Represent the number of apartment units that will be rented in terms of \(x .\)Complex numbers are used to describe current I, voltage \(E,\) and impedance \(Z\)(the opposition to current). These three quantities are related by theequation \(E=I Z, \quad\) which is known as Ohm's Law. Thus, if any two of thesequantities are known, the third can be found. In each exercise, solve theequation \(E=I Z\) for the remaining value. $$E=57+67 i, Z=9+5 i$$Find each product. Write the answer in standard form. $$(3-i)(3+i)(2-6 i)$$Solve each equation or inequality. $$|5 x+2|-2<3$$Write each statement as an absolute value equation or inequality. \(r\) is no less than 1 unit from 29.
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Problem 38 If \(k \neq 0,\) is the equation... [FREE SOLUTION] (2024)

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