Problem 52 - Algebra 2

Question

Simplify each expression. \((3 x+5)-(2 x+3)\)

Step-by-Step Solution

Answer

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

Step 1: Distribute the Negative Sign

First, distribute the negative sign to the terms within the parentheses of the second expression. The expression becomes: \ \((3x + 5) - 1(2x + 3) = 3x + 5 - 2x - 3\)

Step 2: Combine Like Terms

Now, combine the like terms. Group the terms containing \(x\) and the constant terms separately: \ \((3x - 2x) + (5 - 3)\)

Step 3: Simplify the Expression

Simplify each group of like terms: \ - For terms involving \(x\): \(3x - 2x = x\) \ - For constant terms: \(5 - 3 = 2\) \ Thus, the simplified expression is \(x + 2\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental technique in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. They can be combined by adding or subtracting their coefficients. Let's look at how this works.

Imagine the expression we want to simplify is \(3x + 5 - 2x - 3\). The terms are separated into two groups based on their characteristics:

• The terms containing the variable \(x\): \(3x\) and \(-2x\)
• The constant terms: \(5\) and \(-3\)

To combine the like terms with \(x\), you simply add or subtract their coefficients: \(3x - 2x = x\). For the constant terms, you do the same with their values: \(5 - 3 = 2\). So by combining these, the expression simplifies to \(x + 2\). This technique helps keep expressions tidy and simple.

Distributive Property

The distributive property is a useful tool in algebra, allowing us to distribute multiplication over addition or subtraction within an expression. This property is especially helpful when there are parentheses involved.

For example, consider the expression \((3x + 5) - (2x + 3)\). To employ the distributive property, we need to multiply through by the negative sign applied to the second set of terms: \( (3x + 5) - 1(2x + 3) = 3x + 5 - 2x - 3\). Breaking it down:

• The negative sign in front of \(2x + 3\) acts like \(-1\), so each term within the parentheses is multiplied by \(-1\).
• This results in the sign of each term changing from positive to negative, converting \(+2x\) to \(-2x\) and \(+3\) to \(-3\).

Once applied, the expression is ready for the next step in simplification, which is combining like terms.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Understanding how to work with these expressions is crucial in algebra. Let's elucidate these components:

• Variables: These are symbols, often letters like \(x\) or \(y\), that represent numbers. They allow expressions to be generalized and can stand for unknown values.
• Constants: These are known, fixed values like \(5\) or \(-3\) in our expression.
• Coefficients: These are numbers multiplying the variables, such as \(3\) in \(3x\).

When simplifying algebraic expressions like \(3x + 5 - 2x - 3\), recognizing these components helps us apply rules like the distributive property and combining like terms. Understanding each piece allows for clearer, more confident manipulation of expressions, leading to correct solutions and simplified outcomes.