Problem 78 - Elementary Algebra Within Reach

Question

In Exercises \(75-86\), simplify the expression. $$ 7 x-4 x+8+3 x-6 $$

Step-by-Step Solution

Answer

The simplified form of the expression \(7x - 4x + 8 + 3x - 6\) is \(6x + 2\).

Step 1: Combine like terms

We start by combining the terms that involve \(x\). We have \(7x\), \(-4x\), and \(3x\). Adding these together yields: \(7x - 4x + 3x = 6x\).

Step 2: Simplify the constant terms

Now we combine the constant terms, which are 8 and -6. Adding these together, we find: \(8 - 6 = 2\).

Step 3: Final expression

After simplifying the terms involving \(x\) and the constant terms, we combine the results to form a single, simplified expression: \(6x + 2\).

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the most fundamental skills required is combining like terms. Like terms are terms that contain the same variables raised to the same power. The only difference is the coefficients (the numerical part) of these terms. For example, in the expression 7x - 4x + 8 + 3x - 6, the like terms are the ones with the variable x.

To simplify, you only combine the coefficients of these terms while keeping the variable part unchanged. Here’s how it’s done with the given expression:

• Identify like terms: 7x, -4x, and 3x are like terms.
• Add the coefficients: 7 - 4 + 3 gives us 6.
• Combine them with the variable part: 6x.

The process is direct and logical. Repeat the same steps with the constants: 8 and -6, summing them up to 2. The final expression becomes 6x + 2. Combining like terms effectively simplifies the expression to a form that is easier to use in further mathematical operations or applications.

Algebraic Expression Simplification

The simplification of algebraic expressions is a cornerstone of elementary algebra. It involves reducing expressions to their most basic form without changing their value. To simplify an expression, follow these essential rules:

• Combine like terms, as discussed earlier.
• Apply the order of operations (PEMDAS/BODMAS) correctly.
• Use distributive property to eliminate parentheses where necessary.

In the exercise 7x - 4x + 8 + 3x - 6, simplification was achieved by combining like terms. This step reduces complexity and prepares the expression for further manipulation or evaluation. It's also pivotal when solving equations: a simplified equation is far easier to solve. Remember, the goal of simplification is to make the expression as clear and concise as possible while maintaining its original value.

Elementary Algebra

Elementary algebra opens the door to the vast world of mathematical relationships. It includes the fundamentals of working with variables, constants, and the basic operations – addition, subtraction, multiplication, and division. In our exercise, 7x - 4x + 8 + 3x - 6, the application of elementary algebra skills leads to the simplification of algebraic expressions.

Understanding Variables and Constants

Variables (like x in our example) represent unknown quantities, while constants (like 8 and -6) are fixed values. The power of algebra lies in the use of these variables to create general expressions that can solve a multitude of problems.

Why Learn Elementary Algebra?

It is not just a subject learned in school; it's a critical thinking tool. It teaches you to identify patterns, solve problems systematically, and understand the abstract representations that are so crucial in advanced mathematics, science, and many real-world applications. By grasping the essential concept of combining like terms within algebraic expressions, you are laying the foundation for all future mathematical learning and application.