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Chapter 3: Problem 2

Decide if the statement is true or false . The solution of \(x=2 x\) is zero.

Short Answer

Expert verified
The statement is partially correct. While \(x=0\) is a solution to the equation \(x=2x\), it is not the only solution because the equation has infinitely many solutions.

Step by step solution

01

Identify the given equation

The given equation is \(x=2x\).
02

Re-arrange and simplify the equation

To find the solution, it's suggested to isolate x on one side of the equation. Remember, in equation-solving, anything that is done to one side must be done to the other. Therefore, subtract \(x\) from both sides of the equation to get \(0=x-x\). This simplifies to \(0=0\)
03

Analyze the solution

The equation \(0=0\) is a true statement (valid for any \(x\)) which means that there are infinitely many solutions. However, the proposed solution was that \(x=0\), this is true as \(0\) is indeed one of the solutions but the original equation \(x=2x\) has infinitely many solutions, not just \(x=0\)

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Key Concepts

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

Equation Solving
Equation solving is a foundational skill in algebra where the goal is to find the value or values of a variable that makes an equation true. In the process, we manipulate the equation using algebraic operations to isolate the variable. For example, in the equation \(x = 2x\), solving involves getting \(x\) on one side by itself. To achieve this, we subtract \(x\) from both sides, which must be done to maintain the equality of the equation. This leads us to the equation \(0 = x - x\), eventually simplifying to \(0 = 0\), which is always true. The process is essential because it helps us to understand what the equation represents and also reveals the nature of the solutions.
Infinite Solutions
When we find an equation ends in an identity such as \(0=0\), it suggests infinite solutions. This means any value for the variable satisfies the equation. In our example, the transformed equation \(0 = 0\) tells us that no matter what number you choose for \(x\), it will satisfy \(x = 2x\). This occurs because after simplifying, the variables effectively cancel each other out, leaving a statement that is universally true.
  • An equation like this usually represents a dependency where one variable is a multiple of the other or they are equivalent in every context.
  • It's essential to distinguish whether an equation results in infinite solutions or a specific singular solution, as this distinction informs how we understand the relationships within the equation.
Simplifying Equations
Simplifying equations is a step in solving equations where we reduce the equation to its simplest form. This involves combining like terms, removing parentheses by distributing factors, and performing operations that maintain equality but reduce complexity. For \(x = 2x\), simplifying meant re-arranging the equation to \(0 = x - x\), which further reduced to \(0 = 0\).
  • The aim is to make the equation as straightforward as possible to reveal solutions clearly.
  • Simplification can often reveal whether an equation has no solution, one solution, or infinitely many solutions.
  • Always ensure to apply operations equally to both sides to maintain the original balance of the equation. This ensures that no potential solution is lost during the simplification process.

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