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Chapter 2: Problem 9

Identify each equation as an identity or a contradiction. $$6 x-2=6 x+4$$

Short Answer

Expert verified
The given equation is a contradiction

Step by step solution

01

Simplify the equation

Start the problem by simplifying the equation to see if it simplifies to a true statement, a false one, or one that depends on the variable. The given equation is \(6x - 2 = 6x + 4\). In order to simplify, we will bring the similar variables on the left side, and constants on the right side. Subtract \(6x\) from both sides of the equation: \((6x - 6x) = (6x - 6x) + 4 + 2\). The final simplified equation is \(0 = 6\).
02

Identify the type of equation

Next, we will identify whether this equation is an identity, conditional, or contradiction. By observing our simplified equation, \(0 = 6\), we can see that the statement is not true for any value of the variable \(x\). Thus, the equation \(6x - 2 = 6x + 4\) is a contradiction, it is not true for any value of the variable \(x\).

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Key Concepts

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

Understanding Contradictions in Algebraic Equations
In algebra, a **contradiction** occurs when an equation is reduced to a statement that is always false, such as 0 = 6. This means that no matter what value you try to substitute into the variable, the equation can never be true. It's like trying to make a false statement true, which simply isn't possible.
To identify contradictions:
  • Simplify both sides of the equation as much as possible.
  • If your final result is an impossible equation (like saying 1 = 2), then you have a contradiction.
Remember, contradictions tell us that there is no solution to the equation for any value of the variable. This concept is crucial when analyzing whether an equation makes sense and can be solved.
The Role of Identity in Algebraic Equations
An **identity** in mathematics is an equation that is true for every possible value of the variable. It means both sides are fundamentally the same. An example could be something simple like $x+1 = x+1$.
This idea is significant because it ensures that the equation holds no matter what value you plug in for the variable. It鈥檚 like a universal truth for that particular equation:
  • Simplify the equation to see if both sides mirror each other perfectly.
  • If the simplified form results in an equation where both sides are always equal, it is an identity.
Identities are particularly useful because they allow the flexibility of substitution in complex problems, knowing the equation will always hold.
Simplifying Equations Made Simple
**Simplifying equations** is a step-by-step process that helps reveal the basic truth of an equation, whether it鈥檚 a contradiction, identity, or otherwise solvable.
Here鈥檚 how to simplify effectively:
  • Combine like terms. This means terms that have the same variable raised to the same power.
  • Perform basic operations such as addition, subtraction, multiplication, or division on both sides of the equation to isolate terms.
  • Move all terms containing the variable to one side of the equation and constants to the other.
Simplifying clarifies the structure of an equation, making it easier to identify its nature鈥攚hether it's always true, never true, or conditionally true.

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