Problem 41 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ 5(3 a+4 b) \text { , if } a=-2 \text { and } b=2 $$

Step-by-Step Solution

Answer

The value of the expression is 10.

Step 1: Substitute Values of Variables

First, substitute the given values of the variables into the expression. In this case, replace \( a \) with \( -2 \) and \( b \) with \( 2 \) in the expression \( 5(3a + 4b) \). This gives you: \(5(3(-2) + 4(2))\).

Step 2: Simplify Inside the Parentheses

Next, simplify the expression inside the parentheses. Compute each term separately:- For \( 3(-2) \): multiply \( 3 \) and \( -2 \) to get \( -6 \).- For \( 4(2) \): multiply \( 4 \) and \( 2 \) to get \( 8 \).Now, add these results: \( -6 + 8 \) which simplifies to \( 2 \).This gives the expression: \(5(2) \).

Step 3: Multiply the Result

Finally, multiply the number outside the parentheses by the result from inside the parentheses. Compute:\(5 \times 2 = 10\).Thus, the value of the expression is \( 10 \).

Key Concepts

Substitution Method

The substitution method is an invaluable technique when working with algebraic expressions. In our problem, we have certain variables, in this case, \( a \) and \( b \), whose values are provided. The key to using the substitution method is to replace these variables in the expression with their given values. Here’s a step-by-step understanding of this method:

• Identify which values need to be substituted in place of variables. For our exercise, \( a = -2 \) and \( b = 2 \).
• Consider the original expression: \( 5(3a + 4b) \). Substitute \( -2 \) for \( a \) and \( 2 \) for \( b \), transforming it into \( 5(3(-2) + 4(2)) \).

By systematically using this method, you can transform an expression into one that can be simplified and solved efficiently. It helps avoid confusion and mistakes often made when dealing with multiple variables. Remember, careful substitution is the first step towards solving any such expression.

Simplifying Expressions

Simplifying expressions is crucial to bring down complex-looking arithmetic into a simpler form. Once you have substituted variables with their values, the next step is to simplify the expression inside the parentheses. Let's break it down:

• Look at each term independently. For example, \( 3(-2) \) becomes \( -6 \) when multiplied.
• Continue by multiplying \( 4(2) \) to get \( 8 \).

Add these results to simplify further: \( -6 + 8 \) results in \( 2 \). What is happening here is that we are reducing the expression within the parentheses to a single number. It makes the rest of the calculation simpler and more straightforward. Thus, simplifying expressions is all about breaking down the steps and handling them one at a time.

Multiplication of Integers

Multiplying integers is a basic yet essential skill in algebra and involves a few straightforward rules:

• Positive \( \times \) Positive = Positive
• Negative \( \times \) Positive = Negative
• Positive \( \times \) Negative = Negative
• Negative \( \times \) Negative = Positive

In the exercise, once we simplified inside the parentheses, we arrived at \( 5(2) \), indicating multiplication of \( 5 \) by \( 2 \). Since both numbers are positive, the result is a positive \( 10 \). Multiplying integers involves careful consideration of signs and understanding these basic rules can save time and effort. When executed well, this fundamental operation can make solving algebraic expressions efficient and reliable.