Problem 69 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 4 x y-10 x y $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: 4xy - 10xy Answer: -6xy

Step 1: Identify like terms

In the expression given, we can see that the terms "4xy" and "-10xy" are like terms since both involve the product of x and y.

Step 2: Combine like terms

We can combine the like terms by adding their coefficients. So, we have: $$ (4-10)xy $$

Step 3: Simplify the expression

We can simplify the expression by performing the subtraction: $$ (-6)xy $$ So, the simplified expression is: $$ -6xy $$

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the fundamental steps is combining like terms. Like terms are terms that have the same variables raised to the same powers. For instance, in the expression 4xy - 10xy, both terms are considered like terms because they contain the same variables x and y. To combine them, you simply add or subtract their coefficients. Coefficients are the numerical part of the terms; in this example, 4 and -10 are the coefficients.

Here's how you can think of it: imagine you have 4 apples and someone takes away 10 apples, how many apples do you have? In algebraic terms, you have (4 - 10) apples, or in this case, (4 - 10)xy, which simplifies to -6xy. Combining like terms is like basic arithmetic but with a twist of variables!

Simplifying Expressions

The goal of simplifying expressions is to make algebraic expressions as straightforward as possible. This can mean combining like terms, as we have done, or it might involve more complex processes like distributing or factoring, depending on the expression at hand. Simplifying expressions helps us to understand the essential components of an equation or inequality without extra 'noise'.

Think of it as tidying up a room: you want to put away similar items together and throw out what's not needed. Similarly, in algebra, you combine like terms and eliminate any unnecessary parts of the expression to end up with the simplest form, which in the given example was -6xy. Simplifying lets you see the 'cleanest' version of the expression, which makes further calculations, comparisons, or evaluations much easier.

Algebraic Coefficients

In an algebraic expression, the algebraic coefficients are the numerical factors that multiply the variables. They play a crucial role in combining like terms. In our previous example, the coefficients are 4 and -10. When you combine like terms, you essentially just add or subtract these coefficients as you would with regular numbers.

To understand coefficients better, let's compare it to baking cookies. The variable, like x or y, represents the cookie, and the coefficient tells you how many cookies you have. If you have 4x cookies and lose 10x cookies, you'd be left with -6x cookies. Algebraic coefficients determine the quantity of something (in this case, cookies), making them indispensable when simplifying algebraic expressions.

Elementary Algebra

The bread and butter of high school math, elementary algebra is the study of mathematical symbols and the rules for manipulating these symbols. It's essentially a language through which we can represent real-world problems in a simplified, abstract form. Combining like terms, simplifying expressions, and understanding algebraic coefficients are all part of this larger field.

Elementary algebra gives you the building blocks to represent numbers and their relationships in an abstract way. This enables problem-solving that goes far beyond basic arithmetic because it involves unknowns, which we represent as variables. The expression 4xy - 10xy and its simplification to -6xy is a classic example of how elementary algebra operates, turning a visual concept into a neat, mathematical form that's ready for further analysis or application.