Problem 38 - Precalculus Student Solutions Manual 5th

Question

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

Step-by-Step Solution

Answer

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

Step 1 - Understanding the Equation

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

Step 2 - Isolate the Variable

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

Step 3 - Analyze the Result

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

Step 4 - 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

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 circumstance. 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.