Problem 34 - Fundamentals of Mathematics

Question

Find the value of each expression. $$4(3 r+2 s), \text { if } r=4 \text { and } s=1$$

Step-by-Step Solution

Answer

The value of the expression is 56.

Step 1: Substitute Values

First, substitute the given values of \( r \) and \( s \) into the expression. Replace \( r \) with 4 and \( s \) with 1 in the expression \(4(3r+2s)\). This gives us: \[4(3(4) + 2(1))\].

Step 2: Simplify Inside the Parentheses

Next, perform the calculations inside the parentheses. First, calculate \(3 \times 4 = 12\) and \(2 \times 1 = 2\). Substitute these into the expression: \[4(12 + 2)\].

Step 3: Add Inside the Parentheses

Now, add the results inside the parentheses: \(12 + 2 = 14\). This simplifies our expression to \[4(14)\].

Step 4: Multiply Outside the Parentheses

Finally, multiply the result by 4, which is outside the parentheses: \(4 \times 14 = 56\). This gives us the final solution.

Key Concepts

Expression Simplification

Expression simplification is a crucial step in solving algebraic problems. It's all about making the expression easier and more concise while maintaining its original value. Think of it like tidying up a messy room, where your goal is to make everything organized and clear.In our exercise, we started with the expression \(4(3r + 2s)\). The first step involved substituting given values of variables into the expression. We replaced \(r\) with 4 and \(s\) with 1. This substitution converts the abstract representation into something directly calculable:

• Replaced \(r\) by 4
• Replaced \(s\) by 1

After substitution, the expression became \(4(3(4) + 2(1))\). This transformation gives us numbers we can easily work with. Simplification also helps avoid errors in later calculations by breaking down complex expressions into simpler components.

Multiplication

Understanding multiplication in algebra is essential because it allows us to compactly express repeated addition and solve various practical problems.In our example, once we substituted the values, we were left with \[4(3(4) + 2(1))\]First, you multiply the numbers inside the parentheses. In detail:

• Multiply 3 by 4, which gives us 12.
• Multiply 2 by 1, which results in 2.

These operations within the parentheses prepare the numbers for the next step. Multiplication, therefore, is the operation that reduces a longer sum into a simpler arithmetic task. Besides, by mastering multiplication, you're equipped to tackle more complex relationships and expressions efficiently.

Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), dictates the sequence in which calculations should be performed to ensure accurate results in mathematics.In our exercise expression \(4(3r + 2s)\), the parentheses tell us to handle the operations inside them first:

• Calculate \(3(4) + 2(1)\) inside the parentheses.
• Add 12 and 2 to get 14. This sum respects the rule of completing operations inside parentheses before addressing operations outside them.

Once we've simplified inside the parentheses, we move to multiplication outside: multiply 4 by the resulting 14.Ultimately, following the correct order of operations avoids confusion and leads directly to the intended result: 56 in this example. Mastery of this concept ensures that every math problem is tackled correctly and efficiently.