Problem 84 - Algebra for College Students

Question

Simplify each numerical expression. $$5(-1.6)-3(2.7)+5(6.6)$$

Step-by-Step Solution

Answer

The simplified expression is 16.9.

Step 1: Multiply First Pair

First, multiply the numbers in the first term: \(5(-1.6)\). This is a simple multiplication: \(5 \times -1.6 = -8.0\).

Step 2: Multiply Second Pair

Next, handle the second term: \(-3(2.7)\). Multiplying these values results in \(-3 \times 2.7 = -8.1\).

Step 3: Multiply Third Pair

Then, calculate the third term: \(5(6.6)\). The multiplication is \(5 \times 6.6 = 33.0\).

Step 4: Combine All Results

Finally, combine the results from Steps 1, 2, and 3. Calculate \(-8.0 - 8.1 + 33.0\): 1. \(-8.0 - 8.1 = -16.1\) 2. \(-16.1 + 33.0 = 16.9\). Therefore, the simplified expression is 16.9.

Key Concepts

Numerical Expression

A numerical expression is a combination of numbers and operations that represent a specific value. It can include basic arithmetic operations like addition, subtraction, multiplication, and division.
For instance, in our problem, the expression is given as \(5(-1.6) - 3(2.7) + 5(6.6)\).

• This expression combines several operations like multiplication first, and then addition or subtraction.
• Numerical expressions help in representing complex calculations simply.
• The goal here is to evaluate or simplify the expression to get a single numerical answer.

Before solving, always ensure clarity on the order of operations, typically multiplication and division come before addition and subtraction.

Multiplication in Algebra

In algebra, multiplication involves finding the product of two numbers or variables. It is one of the fundamental operations you will often use.
Let's break down how multiplication is handled in our given problem:

• The first term \(5(-1.6)\) is solved by multiplying 5 by -1.6, resulting in -8.0.
• Similarly, the second term \(-3(2.7)\) is calculated as -3 multiplied by 2.7, which equals -8.1.
• In the third term \(5(6.6)\), 5 times 6.6 gives us 33.0.

Each of these multiplications is carried out separately before combining results. In algebra, you should always perform multiplication operations first according to the order of operations (PEMDAS/BODMAS).
This ensures accurate simplification of expressions before engaging in addition or subtraction.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions, although our current expression is purely numerical, the method is very similar.
After performing the necessary multiplications:

• We have results: -8.0, -8.1, and 33.0.
• These are purely numerical and can be combined by performing addition or subtraction as indicated by the expression.

To combine, perform the operations step by step:

• First, add -8.0 and -8.1, which gives -16.1.
• Then, add this result to 33.0.
• Finally, we get 16.9 as the simplified result.

This process is analogous to combining like terms in algebra, where you group and simplify terms with the same variable parts, but here, we simply handle numbers.