Problem 39 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 3 x+2-(5 x+2) $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x\).

Step 1: Distribute Negative Sign

The first step is to distribute the negative sign to every term inside the parentheses. So, the expression becomes \(3x + 2 - 5x - 2\)

Step 2: Combine Like Terms

Combine the similar terms. So the terms \(3x\) and \(-5x\) combine to \(-2x\) and the terms \(2\) and \(-2\) cancel out each other to \(0\). So, the simplified expression will be \(-2x\)

Key Concepts

Distributive Property

The distributive property is a key tool in algebra that allows us to distribute a single term across terms inside parentheses. For example, if we have an expression like \( a(b + c) \), we can apply the distributive property to get \( ab + ac \).

In our exercise, we encounter a slightly different situation, which involves a negative sign. The expression \(3x + 2 - (5x + 2)\) contained a parenthesis with a subtraction sign before it. The negative sign acts as a \( -1 \) multiplier, so when we distribute it over each term inside the parentheses, each term's sign is switched. As a result, \( +5x \) becomes \( -5x \) and \( +2 \) becomes \( -2 \). This step is crucial for setting up the expression for further simplification.

Combine Like Terms

After using the distributive property, the next step in simplifying algebraic expressions is to combine like terms. These are terms that have the exact same variable part, even if the coefficients are different. In our example, \(3x\) and \( -5x \) are like terms because they both contain the variable \(x\).

We combine them by adding their coefficients: \(3\) and \( -5 \), which results in \( -2x \).

Attention to Details

It's essential to pay close attention to the signs of each term when combining them. Similarly, constants without variables are also combined together. In our case, the constants \( +2 \) and \( -2 \) cancel each other out, simplifying to \(0\). This step streamlines our expression, bringing us closer to the most simplified form.

Simplifying Expressions

Simplifying expressions is the overarching goal of the previous steps. The process involves several techniques, including the distributive property and combining like terms, which we've explored. Simplification can also include factoring, dividing terms, and canceling common factors.

The aim is to transform the expression into its simplest or most compact form without changing its value. For instance, after applying the previous steps to our exercise, the expression \(3x + 2 - 5x - 2\) is simplified to \( -2x \).

Why Simplify?

Simplification makes equations easier to read, solve, and allows us to more clearly see the relationships between different algebraic components. It's a fundamental skill in algebra that aids in understanding more complex concepts.

Algebraic Manipulation

Algebraic manipulation encompasses a variety of techniques employed to modify algebraic expressions or equations to make them easier to work with or solve. It involves skills like the distributive property, combining like terms, and the rules for adding, subtracting, multiplying, and dividing algebraic fractions.

Our exercise is a straightforward example of algebraic manipulation, showing how simple techniques can transform an expression. Mastery of algebraic manipulation is not only crucial for solving equations but also for understanding the behavior of functions, analyzing mathematical models, and tackling high-level math problems.

Practice and Patience

Becoming proficient requires practice and a step-by-step approach to ensure each change to the expression is valid and follows mathematical rules.