Question

$$\begin{array}{l}(28.3 x+4.6 x y+37.2 y)-(12.5 x y+44.7 y-6.3 x) \\\\+(3.6 y+2.1 x y-11.6 x) \end{array}$$

Short answer

The simplified expression is 22.0x + (-5.8)xy + (-3.9)y.

Step-by-step solution

Expand the expression

Remove the parentheses by distributing the negative sign to the terms inside the parentheses for the second and third terms to combine like terms. The expression becomes: 28.3x + 4.6xy + 37.2y - 12.5xy - 44.7y + 6.3x + 3.6y + 2.1xy - 11.6x.

Combine like terms

Group and add coefficients of like terms. There are three types of terms to combine: terms with x, terms with xy, and terms with y.

Simplify the expression

Add the coefficients of like terms to simplify the expression to its simplest form: (28.3x + 6.3x - 11.6x) + (4.6xy - 12.5xy + 2.1xy) + (37.2y - 44.7y + 3.6y).

Key concepts

Combining Like Terms

To master algebra, understanding how to combine like terms is essential. Like terms are algebraic expressions that have the same variables raised to the same powers, even if they have different coefficients. For instance, in the given exercise, the terms containing just an 'x', such as 28.3x and -11.6x, are like terms.

To combine them, we focus on adding or subtracting their coefficients (the numerical parts). Imagine these coefficients are like apples in a basket — counting them together gives us the total number of apples. Similarly, we sum the coefficients to simplify the expression. By adding up all the like terms in the exercise, we refine the complex expression into a more manageable equation, setting the foundation for further calculations.

Distributive Property

The distributive property is another cornerstone of algebra that facilitates the simplification of expressions. This property allows us to remove parentheses by distributing a multiplier to each term within the parentheses. Think of it as handing out slices of cake evenly to friends at a party. Each slice (term inside the parentheses) must be multiplied by the number of friends (the number outside the parentheses).

In the original exercise, we observe this property in action when we eliminate the parentheses by distributing the negative sign to each term inside. By doing so, we ensure every term is accounted for correctly in the next steps of simplification. Understanding the distributive property is crucial for tackling more complex algebraic equations efficiently.

Algebraic Manipulation

Finally, algebraic manipulation encompasses various techniques, including the two previously mentioned concepts, to rearrange and simplify algebraic expressions. This general term refers to the skillful handling of expressions to make them easier to understand or to solve. In our exercise, we manipulate the algebraic terms by first using the distributive property to eliminate brackets and then combining like terms to produce a cleaner, more simplified version of the original complex expression.

Think of algebraic manipulation as a toolkit. Each 'tool'—whether it's the distributive property, combining like terms, or other strategies like factoring — serves a unique purpose in simplifying and solving algebraic challenges. It requires practice, but with time, algebraic manipulation will become an intuitive process, making algebra less daunting and more approachable.