Problem 9 - Elementary Algebra

Question

Solve each of the equations. $$x=3.36-0.12 x$$

Step-by-Step Solution

Answer

\( x = 3 \)

Step 1: Move all terms involving x to one side

Our goal is to isolate the variable \( x \) on one side of the equation. Start by adding \( 0.12x \) to both sides of the equation, resulting in: \[ x + 0.12x = 3.36 \] Thus, the equation becomes: \[ 1.12x = 3.36 \].

Step 2: Solve for x

To find \( x \), we need to divide both sides of the equation by 1.12. This will isolate \( x \): \[ x = \frac{3.36}{1.12} \] Calculate the division to get the value of \( x \): \[ x = 3 \].

Key Concepts

Solving Equations

Solving equations is a fundamental part of algebra that requires you to find the value of a variable that makes an equation true. In this exercise, the goal is to solve an equation where the variable, represented by \( x \), appears on both sides. The first step is to simplify and rearrange the terms so that all the terms involving \( x \) are on one side of the equation. This process often involves:

• Adding or subtracting terms to both sides
• Combining like terms to simplify the equation

In our example, we add \( 0.12x \) to both sides, effectively moving the \( x \) terms together. This operation leads to a new, simplified equation, \( 1.12x = 3.36 \). At this point, the problem is simpler as it consists of isolating the variable, which is a crucial next step.

Variable Isolation

Variable isolation is a key technique in solving equations, especially linear equations like the one in this exercise. The importance of isolating the variable lies in its ability to lead us to the solution directly. Here's how it works:Once you have gathered all the terms with \( x \) on one side, you need to ensure that \( x \) stands alone. In our worked-out problem, we reached the equation \( 1.12x = 3.36 \). To isolate \( x \):

• Divide both sides of the equation by the coefficient of \( x \), which in this case is 1.12.

This yields \( x = \frac{3.36}{1.12} \), a straightforward division problem.The process of leaving \( x \) alone on one side simplifies the equation and makes it easy to calculate the solution. Once the variable is isolated and simplified, you can solve it quickly, getting the precise value of \( x \).

Basic Algebra

Awareness and understanding of basic algebra principles are crucial when dealing with equations. Concepts like collecting like terms, balancing equations, and performing arithmetic operations accurately, form the backbone of solving problems. These fundamentals mean acting upon the properties of equality consistently, which keeps the equation balanced.In our equation \( x = 3.36 - 0.12x \):

• Recognize \( 0.12x \) as a term involving the variable, needing a move to the other side
• Combine like terms to reduce complexity
• Use division to finalize the isolation

By consistently applying basic algebra rules such as these, solving equations becomes a series of logical steps. This systematic approach helps remove confusion and enables even complex equations to be solved methodically. Mastering these core aspects of algebra sets a strong foundation for tackling more advanced math problems.