Problem 66 - Elementary and Intermediate Algebra

Question

List the terms in each expression. $$ 3 x y+20+\frac{4 a}{b} $$

Step-by-Step Solution

Answer

The terms are \[3xy\], \[20\], and \[\frac{4a}{b}\].

Step 1: Identify Each Term

A term in an expression is a single part of the expression that can be a number, a variable, or the product of numbers and variables. Look at the given expression: \[3xy + 20 + \frac{4a}{b}\] The terms in this expression are separated by the plus signs. Identify each part separated by the plus signs.

Step 2: Write Down the First Term

The first term in our expression is \[ 3xy\] since this is the part of the expression that comes before the first plus sign.

Step 3: Write Down the Second Term

Next, identify the part of the expression that comes after the first plus sign but before the next plus sign, which is \[20\].

Step 4: Write Down the Third Term

Finally, identify the part of the expression that comes after the second plus sign, which is \[\frac{4a}{b}\].

Key Concepts

Terms

In algebra, a term is a single mathematical expression that can be a single number, a variable, or a combination of both multiplied together. Terms are the building blocks of algebraic expressions. For example, in the expression \(3xy + 20 + \frac{4a}{b}\), there are three terms:

• \(3xy\)
• \(20\)
• \(\frac{4a}{b}\)

Each term is separated by plus or minus signs. Understanding how to identify and work with terms is critical for simplifying expressions and solving algebraic equations.

Variables

Variables are symbols that represent unknown or changeable values. They are usually denoted by letters such as \(x\), \(y\), or \(a\). In the expression \(3xy + 20 + \frac{4a}{b}\), the variables are:

• \(x\)
• \(y\)
• \(a\)
• \(b\) (appearing in the denominator)

Variables allow us to generalize mathematical problems and form equations to solve for their unknown values. They play a crucial role in algebra since they enable the representation of mathematical relationships and functions.

Constants

Constants are fixed values that do not change. Unlike variables, constants have a specific value. In the expression \(3xy + 20 + \frac{4a}{b}\), the constant is:

• \(20\)

Numbers by themselves are always constants. They provide a reference point in algebraic expressions. Even when combined with variables, constants help define the nature of the expression or equation.

Fractions

A fraction is a way of expressing a number that is not whole. Fractions consist of a numerator (the top part) and a denominator (the bottom part). In the expression \(3xy + 20 + \frac{4a}{b}\), \(\frac{4a}{b}\)\ is a fraction where:

• \(4a\) is the numerator
• \(b\) is the denominator

Fractions can involve both constants and variables. Working with fractions in algebra typically involves understanding how to add, subtract, multiply, and divide them. Recognizing and simplifying fractions is essential for solving many algebraic problems.