Problem 2 - Elementary Algebra

Question

Solve each of the equations. $$x-0.15=0.42$$

Step-by-Step Solution

Answer

The solution is \(x = 0.57\).

Step 1: Understand the Equation

First, let's examine the given equation: \(x - 0.15 = 0.42\). Our goal is to solve for \(x\) by isolating it on one side of the equation.

Step 2: Add 0.15 to Both Sides

To isolate \(x\), we need to eliminate \(-0.15\) on the left side. We do this by adding \(0.15\) to both sides of the equation. This gives us: \[x - 0.15 + 0.15 = 0.42 + 0.15\].

Step 3: Simplify the Equation

On the left side, the \(-0.15\) and \(+0.15\) cancel out, leaving us with \(x\). On the right side, we simplify \(0.42 + 0.15\), which equals \(0.57\). Our equation now is: \[x = 0.57\].

Key Concepts

Isolation of Variables

When solving linear equations, isolating the variable is a crucial step. The term "isolating a variable" means arranging the equation so that the variable you are solving for is on one side by itself. This process often involves performing operations that cancel out other terms or numbers on the same side as the variable.
In the given equation, we started with:

• \( x - 0.15 = 0.42 \)

Our task is to have \(x\) all alone on the left side. We need to remove \(-0.15\) by performing the opposite operation, which is adding \(0.15\). By adding \(0.15\) to both sides, we ensure that we do the same thing on both sides of the equation, maintaining the equality. This gives us the updated equation where \(x\) is isolated:

• \( x = 0.57 \)

Remember, the goal is to manipulate the equation until the variable stands by itself. This concept is foundational in algebra and necessary for solving many types of problems.

Step by Step Algebra

Breaking down algebra problems into clear, manageable steps makes them much easier to tackle. Each step should move you closer to isolating the variable while maintaining the balance of the equation.
Begin with understanding the equation and the objective. In our case, we have:

• \( x - 0.15 = 0.42 \)

Decide what operations will help isolate \(x\). Then, follow through with the operation:

• Add \(0.15\) to both sides resulting in \( x = 0.57 \).

This step-by-step method ensures clarity and consistency, helping prevent mistakes. Each step should be validated; once you've achieved the isolation of the variable, your work is complete. This methodical approach can be applied to any linear equation, turning complex algebra into systematic operations.

Simple Arithmetic Operations

Algebra often involves basic arithmetic operations such as addition, subtraction, multiplication, and division. These are the tools used to manipulate and solve equations.
In our exercise, the operation used was addition:

• Add \(0.15\) to both sides of the equation to solve for \(x\).

Once you identify the operation needed (in this case, to counteract subtraction), apply it to both sides:

• \( x - 0.15 + 0.15 = 0.42 + 0.15 \)

The subtraction and addition of \(0.15\) cancel each other on the left, simplifying to:

• \( x = 0.57 \)

Arithmetic operations lay the groundwork for solving equations. By accurately performing these operations, we can effectively isolate the variable and find its value. They are essential for maintaining the balance and integrity of the equation throughout the solving process.