Problem 52 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 8 l-(3 l-7) $$

Step-by-Step Solution

Answer

The simplified expression is \(5l + 7\).

Step 1: Distribute Negative Sign to Parentheses

First deal with the negative sign in front of the parentheses. In essence, this negative sign gets distributed to each term inside the parentheses. Thus the expression changes to: \(8l - 3l + 7\)

Step 2: Combine Like Terms

Combine the like terms (i.e terms with similar variables). The 'l' terms are like terms. Proceed by subtracting 3l from 8l which gives: \(5l + 7\)

Step 3: Write the Final Simplified Expression

No further simplification can be done to the expression. So, the final simplified expression remains as: \(5l + 7\)

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions can sound intimidating at first, but it's all about making the expression easier to work with. Think of it as tidying up a messy room - gathering all the socks in one place or organizing your books by size. Here are some important tips:

• Pay close attention to operations like addition, subtraction, multiplication, and division.
• Remove grouping symbols like parentheses or brackets by performing the operations indicated.
• Always aim to express the equation in the simplest form possible.

In the given problem, the expression originally involved a subtraction within parentheses: \(8l-(3l-7)\). By simplifying, we ensure the expression is easier to read and understand, confirming its true value or meaning.

Distributive Property

The distributive property is a fundamental concept in algebra that lets you multiply a number or variable by each part within a set of parentheses. Think of it like handing out candies evenly to a group of children. Here's how you engage with it:

• If you see a sign (like a negative or a number) outside the parentheses, it must be distributed to each term inside.
• Applies to both multiplication and addition/subtraction, leading to crucial transformations in expressions.

In the problem \(8l-(3l-7)\), the negative sign before \((3l-7)\) means you need to multiply \(-1\) by both \(3l\) and \(-7\). This removes the grouping symbol to result in \(8l - 3l + 7\). It makes operations on the terms inside the parentheses possible, streamlining simplifying further.

Combining Like Terms

Once you remove grouping symbols using the distributive property, you often find yourself with several similar terms. These are what we call "like terms". Think of combining like terms like separating apples from oranges in a basket:

• Look for terms that have the same variable raised to the same power, like \(3l\) and \(8l\).
• Add or subtract their coefficients directly, as the variables themselves remain constant in the equation.
• Remember, only like terms can be combined for simplification.

In our exercise example, \(8l - 3l\) are like terms because they share the variable \(l\). By subtracting \(3l\) from \(8l\), we get \(5l\), reducing the expression to \(5l + 7\), beautifully simplified and easy to understand.