Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the given equation. $$ 2 x=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(2x = 0\) is \(x = 0\).

Step 1: Identify the equation

We are given the equation \(2x = 0\). Our goal is to find the value of x that makes this equation true.

Step 2: Divide both sides by 2

Divide both sides of the equation by 2 to isolate the variable x: \(\frac{2x}{2} = \frac{0}{2}\)

Step 3: Simplify and find the value of x

Simplify the equation as follows: \(x = 0\) The value of x that makes the equation true is x = 0.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. In algebra, letters and other symbols are used to represent numbers and quantities in formulas and equations. Understanding basic algebra is crucial as it lays the foundation for higher mathematical concepts and a wide range of real-world problem solving.

For students starting with algebra, it can be seen as an extension of arithmetic, where unknown values are represented by letters such as x or y. These unknowns can be anything from simple integers to complex numbers. One of the primary goals in algebra is to find these unknown values, which are often referred to as solving for the variable.

Equation Solving Step by Step

The process of solving an equation involves a series of steps to find the value(s) of the variable(s) that satisfy the equation. The basic steps can be generalized as follows:

• Understand the Problem: Carefully read the equation to understand what is being asked. Identify the variables and constants.
• Simplify the Equation: If necessary, simplify the equation by consolidating like terms and removing parentheses.
• Isolate the Variable: Use algebraic operations to get the variable on one side of the equation and constants on the other.
• Perform Operations: Carry out any required operations to solve for the variable, which can include addition, subtraction, multiplication, division, or factoring.
• Check Your Solution: Substitute the solution back into the original equation to ensure it makes the equation true.

Effective equation solving not only involves performing the correct steps but also requires understanding why each step is taken to build a solid foundation in algebra.

Isolate Variable

To isolate the variable means to get the variable by itself on one side of the equation. This is a fundamental skill in algebra which allows us to find the value of the variable. Depending on the complexity of the equation, isolating the variable can involve a combination of adding, subtracting, multiplying, or dividing both sides by the same number.

Let's look at a simple example such as the equation from the provided exercise, where we have the equation \(2x = 0\). To isolate the variable x:

• We divide both sides of the equation by 2, the coefficient of x.
• This gives us \(\frac{2x}{2} = \frac{0}{2}\), which simplifies to x = 0.

This process demonstrates the importance of performing the same operation on both sides of the equation to maintain its balance, which is a key principle in equation solving.