Question

Combine as indicated and simplify. $$22.6 x y+38.3 x y-33.9 x y+58.2 x y$$

Short answer

The simplified expression is 85.2xy

Step-by-step solution

Identify like terms

Combine the terms that have the exact same variables to the same power, which in this case are all the terms because they all contain the variable 'xy'.

Combine the coefficients

Add or subtract the coefficients (numerical parts) of these terms: 22.6, 38.3, -33.9, and 58.2.

Simplify the expression

Calculate the sum of the coefficients and multiply the result by the variable part 'xy' to get the simplified expression.

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that contains numbers, variables (like x or y), and operations (such as addition, subtraction, multiplication, and division). In a way, it's like a recipe that tells you how to mix these different ingredients to create something new. Just like baking a cake requires a certain amount of each ingredient, combining xy terms in an algebraic expression requires careful attention to the quantities, which in math we call coefficients.

For instance, in the given expression 22.6xy + 38.3xy - 33.9xy + 58.2xy, x and y are our variables, which can be thought of as placeholders for numbers. The operations are simple: addition and subtraction. Each term's coefficient tells us how much of that 'ingredient' we have. When we combine these, we're gathering like terms to see the total amount of 'xy' we're working with.

Simplifying Expressions

Simplifying an expression is a bit like cleaning up after that baking we talked about: you want a tidy, neat kitchen, or in this case, the simplest form of an expression. To do this, we often combine like terms. Like terms are terms in an expression that have the exact same variables raised to the same power. Think of them as matching socks: they're easy to pair up.

When we simplify the expression 22.6xy + 38.3xy - 33.9xy + 58.2xy, we're pairing up all the 'xy' terms, just like we'd match socks with the same pattern. Once we've got them together, we can just add or subtract their coefficients to find out how many 'xy' socks—or, in mathematical terms, how much of the variable we actually have.

Coefficients

In algebra, the coefficient is the number in front of the variable that acts like a multiplier. For example, in the term 22.6xy, 22.6 is the coefficient. If we think of the 'xy' as a container, the coefficient tells us how much of something is in that container.

Adding and Subtracting Coefficients

To combine terms with the same variables, we add or subtract their coefficients. It's like saying if we have 22.6 buckets of 'xy' and someone gives us 38.3 more buckets, but then takes away 33.9 buckets, and we find another 58.2 buckets, how much 'xy' do we end up with? Simply add up these numbers, respecting the signs (+ or -), and that's the total coefficient for the 'xy' terms in our simplified expression.