Problem 99 - Introductory Algebra for College Students

Question

Simplify: \(\quad 7 x-(3 x-5)\)

Step-by-Step Solution

Answer

The simplified form of the given expression is \(4x + 5\).

Step 1: Distribute the negative sign

The expression provided is \(7 x - (3 x - 5)\). Before simplifying further, let's distribute the negative sign over the parentheses: \(7 x - 3 x + 5\).

Step 2: Combine like terms

After distributing, combine like terms to simplify the expression further. We combine \(7x\) and \(-3x\) terms together: \(4x + 5\).

Key Concepts

Distributive Property

Understanding the distributive property is crucial when simplifying algebraic expressions. It's a bridge that connects multiplication and addition or subtraction. Essentially, it allows us to remove parentheses by distributing a multiplier to each term within the parentheses. For instance, when we have the expression \( a \times (b + c) \), the distributive property lets us rewrite it as \( a \times b + a \times c \).

Now, let's relate this to our original problem: \( 7x - (3x - 5) \). We need to distribute the negative sign across the items inside the parentheses. It's like saying we have \( -1 \times 3x \) (which is \( -3x \)) and \( -1 \times -5 \) (which is \( +5 \)) because a negative times a negative becomes a positive. Therefore, our expression becomes \( 7x - 3x + 5 \), setting the stage for further simplification.

Combining Like Terms

Once we've distributed properly, the next step is to look for 'like terms.' Like terms are terms that have the same variable raised to the same power. By combining them, we simplify the expression into a more manageable form. It's like consolidating items in an inventory; instead of saying you have 3 apples here and 2 apples there, you'd say you have 5 apples total.

In our exercise, after distributing, we have the terms \( 7x \) and \( -3x \). They are like terms because they both have the variable \( x \) raised to the first power. Combining them is straightforward: \( 7x - 3x \) becomes \( 4x \). Now, we'll have the expression \( 4x + 5 \), where the numbers are piled together neatly, making the expression cleaner and easier to understand.

Algebraic Simplification

Algebraic simplification refers to the process of reducing an expression to its simplest form. This involves the use of several algebraic techniques including the distributive property and combining like terms, as seen in our example problem. The goal is to make the expression as straightforward as possible, sometimes to facilitate further algebraic operations or to evaluate the expression.

After utilizing the distributive property and combining like terms, the expression \( 7x - (3x - 5) \) has already been simplified to \( 4x + 5 \). This expression can't be combined any further because \( 4x \) and \( 5 \) are not like terms. The term \( 4x \) includes the variable \( x \), while \( 5 \) is a constant term. Hence, we've achieved algebraic simplification, arriving at an expression that is easy to work with, whether for graphing, solving equations, or applying in real-world problem scenarios.