Question

Combine as indicated and simplify. $$47.2 x y+12.6 x y+38.3 x y-39.9 x y+18.2 x y$$

Short answer

The simplified form of the expression is 76.4xy.

Step-by-step solution

Identify Like Terms

The given terms are all alike as they all contain the variables x and y. All coefficients can be combined because they share the same variables.

Combine the Coefficients

Add the coefficients of the like terms together. This includes adding 47.2, 12.6, 38.3, -39.9, and 18.2, while keeping the xy attached to each term.

Perform the Addition and Subtraction

Add the positive coefficients and subtract the negative coefficients: (47.2 + 12.6 + 38.3 + 18.2) - 39.9.

Calculate the Sum

Perform the arithmetic operations to find the sum of the coefficients: 47.2 + 12.6 + 38.3 + 18.2 = 116.3 and subtract 39.9 from this sum to get the final coefficient for xy.

Write the Final Expression

Attach the final coefficient to xy to express the simplified form of the given expression. The final coefficient is the result of the calculation from the previous step.

Key concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (such as addition, subtraction, multiplication, or division). In the context of our original exercise, the algebraic expression consists of terms, each featuring the variables 'x' and 'y' along with a numerical coefficient. When we encounter an algebraic expression like \(47.2xy + 12.6xy + 38.3xy - 39.9xy + 18.2xy\), it's critical to understand that each term is a part of a larger mathematical phrase that represents a specific quantity.

These expressions can be thought of like sentences in a language, where variables and coefficients are words that provide meaning to the numerical narrative. The variables (x and y, in this case) can represent unknown values or quantities that can change, while the coefficients (like 47.2 or -39.9) are specific numbers that modify the variables, similar to adjectives changing nouns in a sentence.

Simplifying Expressions

Simplifying expressions is a fundamental process in algebra where you reduce an expression to its most basic form. This involves combining like terms, which are terms in an expression that have the exact same variable factors, regardless of their coefficients. In our exercise, all terms are considered like terms because they all contain the variable pair \(xy\).

The goal is to combine the coefficients of these like terms to simplify the expression. For instance, adding \(47.2\) and \(12.6\) because they're both attached to \(xy\), while being mindful of the operation signs. The simplified form makes the expression cleaner and often easier to work with in equations or further algebraic manipulations. It is important to perform the arithmetic operations carefully to ensure accuracy in the end result.

Polynomials

Polynomials are algebraic expressions that include sums and products of variables and coefficients. The degrees of a polynomial are determined by the highest exponent of its variable terms. While our original exercise doesn't demonstrate varying degrees (since all terms are first-degree with respect to both \(x\) and \(y\)), it's important to recognize it as a simple polynomial.

Polynomials are classified by the number of terms they have: monomial (1 term), binomial (2 terms), trinomial (3 terms), and so forth. Simplifying polynomial expressions involves combining like terms and is an essential step in managing more complex algebraic operations such as factoring, finding zeroes, and graphing polynomial functions. When working with polynomials, always remember to align like terms before combining to streamline the simplification process.