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Chapter 3: Problem 4

Determine whether the given equation is an identity or a contradiction. $$4 x-1=4(x-1)$$

Short Answer

Expert verified
The equation is a contradiction because $$-1 鈮� -4$$.

Step by step solution

01

Expand the right-hand side

Expand the equation on the right-hand side: $$4 x-1 = 4(x-1)$$ $$= 4x - 4$$
02

Write the expanded equation

Rewrite the equation with the expanded form of the right-hand side:$$4x - 1 = 4x - 4$$
03

Eliminate the common term

Subtract 4x from both sides of the equation:$$4x - 1 - 4x = 4x - 4 - 4x$$ $$-1 = -4$$
04

Analyze the result

Observe the result $$-1 = -4$$ This is a false statement, indicating a contradiction.

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Key Concepts

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

solving linear equations
To solve a linear equation, you need to find the value of the variable that makes the equation true. For example, with the equation: \(4x-1=4(x-1)\), we must determine if the equation holds true or if it reveals a special case like an identity or contradiction.

Start by performing operations to simplify and isolate the variable. Here are some essential steps:
  • Expand any expressions, if necessary.
  • Combine like terms on both sides of the equation.
  • Move terms involving the variable to one side of the equation.
  • Move constant terms (numbers without the variable) to the other side.
For our example, we start by expanding the right-hand side. This gives us: \(4x - 1 = 4x - 4\). Next, localize terms involving the variable on one side, which simplifies the equation.
expanding algebraic expressions
Expanding algebraic expressions involves removing parentheses and applying the distributive property. This helps to simplify complex equations.

In our example: \(4x - 1 = 4(x - 1)\), we need to expand \(4(x - 1)\). Use the distributive property, which states: \(a(b + c) = ab + ac\). So, apply this to get: \(4x - 4\).
Rewrite the original equation incorporating this expansion: \(4x - 1 = 4x - 4\).
Now, the equation is simplified and ready for further analysis. Expanding expressions is crucial for solving equations accurately.
algebraic contradictions
An algebraic contradiction occurs when an equation simplifies to a false statement, indicating that no value of the variable will satisfy the equation.

For our example: \(4x - 1 = 4x - 4\), we subtracted \(4x\) from both sides to isolate the constant terms. This resulted in: \(-1 = -4\), which is clearly false.
Therefore, the equation is a contradiction. Remember, contradictions signify that the initial equation has no solution. Identifying contradictions is crucial in algebra as it helps in understanding the nature of equations you're working with.

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