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Chapter 2: Problem 25

Solve each equation. $$ 3 x=0 $$

Short Answer

Expert verified
The solution to the equation is \(x = 0\).

Step by step solution

01

Understand the Equation

The given equation is a simple linear equation: \(3x = 0\). This means we have a constant, 3, multiplied by the variable \(x\), and it is equal to 0.
02

Isolate the Variable

To solve for \(x\), we need to isolate it. Since \(x\) is multiplied by 3, we can divide both sides of the equation by 3 to get \(x\) alone. Our equation becomes: \( \frac{3x}{3} = \frac{0}{3} \).
03

Solve for \(x\)

When we divide both sides of the equation, we get \(x = 0\). Therefore, \(x\) is solved.

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Key Concepts

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

Equation Solving
The process of solving an equation involves finding the value of the variable that makes the equation true. In the equation \(3x = 0\), you have a linear equation, which is the simplest form of equations. Linear equations are equations of the first degree, which means the variable is raised to the power of one.

Here's how to approach equation solving in general:
  • Identify the equation you need to solve. In this case, \(3x = 0\).
  • Understand the relationship between the variables and constants in the equation.
  • Perform operations to simplify the equation. The goal is to maintain equality while making it easier to find the solution.
Breaking down these steps helps you focus on what the equation expresses and sets the stage for finding the solution.
Variable Isolation
Variable isolation is all about getting the variable alone on one side of the equation. In linear equations, this usually involves either adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

In the example \(3x = 0\), you want to isolate \(x\):
  • The variable \(x\) is multiplied by 3.
  • To undo this multiplication, divide both sides by 3.
This operation transforms the equation to \(x = \frac{0}{3}\), simplifying to \(x = 0\). After the division, the variable is isolated, and you have found the solution. This concept is crucial because isolating the variable is often the key to solving any equation.
Basic Algebra
Basic algebra involves understanding how to work with numbers and variables in equations. It's the foundation of much of mathematics, and it starts with operations and principles that you can apply to solve various equations.

Here are a few fundamental strategies of basic algebra used in our problem:
  • Multiplication and Division: These operations are used to manipulate equations so you can isolate variables. In our equation, we divided by 3 to isolate \(x\).
  • Simplification: By simplifying both sides of the equation (e.g., dividing \(0\) by 3), you help to see more clearly what the equation tells you about the variable.
By mastering these basic concepts, you can confidently tackle linear equations and make sense of more complex algebraic expressions as you progress in mathematics. Remember, practicing these fundamentals builds a strong foundation for all math learning.

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