Problem 47 - Elementary Algebra

Question

Solve. $$ 4.5 x-2.3=6.7 $$

Step-by-Step Solution

Answer

x = 2.0

Step 1: Move the Constant to the Other Side

Start by adding 2.3 to both sides of the equation to eliminate the constant on the left side: \[ 4.5x - 2.3 + 2.3 = 6.7 + 2.3 \] This simplifies to:\[ 4.5x = 9.0 \]

Step 2: Isolate the Variable

Next, divide both sides of the equation by 4.5 to solve for \( x \):\[ \frac{4.5x}{4.5} = \frac{9.0}{4.5} \] This simplifies to:\[ x = 2.0 \]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. An equation is like a balance scale, where both sides need to be equal. Our goal is to find the value of unknowns, usually represented by letters like \( x \). Here's a simple process to follow:

• Identify what you're solving for, known as the variable.
• Plan how to isolate this variable to get it alone on one side of the equation.
• Perform the same operations on both sides to maintain balance. Think of it like maintaining harmony in the equation world.

In our example, the equation is \( 4.5x - 2.3 = 6.7 \). We aim to solve for \( x \) by moving numbers around and performing operations until \( x \) stands alone. Understanding that equations need to stay balanced is key to solving them correctly.

Variable Isolation

Variable isolation is all about getting your variable, such as \( x \), all by itself. It involves untangling the variable from the rest of the numbers in the equation. Here's how we achieve that:

• First, remove any constants on the same side as the variable by adding or subtracting them.
• Then, handle any coefficients attached to the variable by multiplying or dividing.

In our given equation, \( 4.5x - 2.3 = 6.7 \), the constant \(-2.3\) is moved to the other side with addition. This leaves \( 4.5x = 9.0 \). Next, we divide by the coefficient \( 4.5 \) to solve for \( x \). The goal is to always have \( x = \) something, making the solution clear and straightforward.

Elementary Algebra

Elementary algebra is the starting point for all algebraic concepts and skills. It revolves around manipulating numbers and variables to solve problems. Here's what you should know:

• Understanding the structure of algebraic expressions and equations.
• Learning how to perform arithmetic operations like addition, subtraction, multiplication, and division.
• Recognizing patterns and simplifying expressions to make solving easier.

The problem we solved, \( 4.5x - 2.3 = 6.7 \), is a classic example of using basic algebraic principles. By methodically performing operations, we simplified the equation to find \( x = 2.0 \). Grasping these foundational skills is vital for more advanced algebraic challenges.