Problem 110 - Introductory Algebra for College Students

Question

Write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. 14 added to the product of 3 and \(-15\)

Step-by-Step Solution

Answer

The simplified value of the expression is -31.

Step 1: Translate Phrase to Equation

The phrase '14 added to the product of 3 and -15' translates into this equation: 14 + 3 * -15.

Step 2: Perform Multiplication

Using the rules of BIDMAS / PEMDAS, multiplication is done before addition, so next is to perform the multiplication: 3 * -15 = -45.

Step 3: Perform Addition

Now, add 14 to -45 to get the final result: 14 + -45 = -31.

Key Concepts

Numerical Expressions

A numerical expression is a mathematical phrase that includes numbers and operational symbols, but no equals sign. It functions like a sentence in math. You use it to represent a number or calculation in a straightforward form. In the phrase from our exercise, "14 added to the product of 3 and -15," the task is to translate those words into a numerical expression.

You simply replace words with numbers and symbols:

• "14 added to" becomes "+ 14"
• "product of 3 and -15" becomes "3 * -15"

Putting it all together gives: \[14 + 3 \times -15\].
Crafting a numerical expression is like writing a sentence, so make sure you understand the operation words you encounter!

Simplifying Expressions

Simplifying an expression means finding its simplest form. It involves performing all possible operations to reduce the expression to a single number. In our example, the initial expression is \[14 + 3 \times -15\].

Following the order of operations, which you may know as BIDMAS or PEMDAS, means:

• Brackets, Indices, Divisions/Multiplications, Additions/Subtractions

For the provided problem, you first tackle the multiplication, because it comes before addition in the order:
\[3 \times -15 = -45\].

Now the expression is just:

• \[14 + (-45)\]

Finally, perform that addition to reach the simplest form:
\[14 + (-45) = -31\].
Simplifying doesn’t change the value, it just makes it easier to understand.

Basic Operations

Basic operations in mathematics are fundamental skills every learner needs to master. They consist of addition, subtraction, multiplication, and division. In the exercise, we mostly see addition and multiplication at work. Let’s break these down a bit more.

• **Addition:** Combining two or more numbers to get their total. In our case, you eventually add 14 to the product of the others.
• **Multiplication:** A method of adding a number to itself a certain number of times. Here, the product '3 times -15' tells you to multiply -15 by 3.

These operations follow specific rules. The order of operations (like BIDMAS/PEMDAS) ensures you perform these in the right order:
1. Multiply or divide before you add or subtract.
2. Always move left to right within expressions.

Understanding these basic operations and how to prioritize them is crucial in simplifying numerical expressions and solving equations correctly.