Problem 49 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 6(2 s-1)+s+4 $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(13s - 2\).

Step 1: Distribute the multiplication

The first step is to distribute the multiplication inside the brackets. The expression 6(2s - 1) results in \(12 s - 6\) when you apply distribution.

Step 2: Combine the expression

Combine the distributed expression with the rest of the expression given in the problem. So the expression becomes \(12s - 6 + s + 4\).

Step 3: Combine like terms

12s and s are like terms, so combine them. Similarly, -6 and 4 are constants, so combine them as well. The final simplified expression after combining like terms will be \(13s - 2\).

Key Concepts

Distribution of Multiplication

Grasping the concept of distribution in algebra can be quite a game-changer for students. When you come across an expression like 6(2s - 1), it's essential to know how to distribute the multiplication over both 2s and -1. This step might seem tricky at first, but it's like sharing cookies evenly amongst your friends.

Picture the number 6 as a generous giver. It multiplies itself with each term inside the parentheses, sharing its value. So, 6 times 2s gives us 12s, and 6 times -1 gives us -6. Voila! You've just distributed the multiplication and your expression unwraps into \(12s - 6\). It's almost like magic, but with numbers!

Combining Like Terms

After you've distributed the multiplication, you might notice your expression looks a bit cluttered with several terms hanging around. It's time to bring order to the chaos by combining like terms. Imagine like terms as siblings – they share the same variable and power, making them a perfect match to come together.

For example, in the expression \(12s - 6 + s + 4\), notice that \(12s\) and \(s\) are like terms since they share the same variable, s. They need to stick together. Think of adding these as tallying points in a game - \(12s + s\) equals \(13s\). Similarly, for the numbers without variables, or the 'constants' as they're called, \( -6\) and \(4\) are a pair. When they come together, they give \( -2\). Thus, the clutter clears, and you are left with a neat \(13s - 2\).

Symbols of Grouping

Symbols of grouping are like the instructions for solving a puzzle. They tell us in what order to tackle the problem. In algebra, these symbols can be parentheses \(( )\), brackets \[ \] or braces \{ }\. Always address what's inside these symbols first - it's like opening a present, you must remove the wrapping before getting to the good stuff.

When simplifying expressions, remove these symbols carefully. Make sure you've followed the other algebraic 'gift-wrapping' rules first like distributing the multiplication over the terms inside. Once done, like peeling off a sticker, you gently remove the symbols of grouping, and your algebraic expression is free to combine like terms and become its simplest self.