Question

Combine as indicated and simplify. $$-9 x y+2 x y$$

Short answer

\(-9 x y + 2 x y = -7 x y\)

Step-by-step solution

Combine like terms

Identify and combine like terms. Here, both terms are like terms because they have the same variables with the same exponents, namely, \(x\text{ and }y\).

Simplify the expression

To combine the like terms, simply add the coefficients of the like terms together. In this case, \( -9 \text{ and } 2 \) are the coefficients of the like terms \(x y\), so you will calculate \( -9 + 2 \) to get the simplified form.

Key concepts

Simplify Expressions

When working with algebraic expressions, the goal is often to 'simplify' them. This can mean combining like terms, reducing fractions, or applying distributive properties to remove parentheses. Simplifying an expression makes it more compact and easier to understand while keeping its original value unchanged.

For example, if you encounter an expression like \( -9xy + 2xy \), simplifying involves finding terms that can be combined because they represent the same quantity or item, known as 'like terms.' In this expression, both terms have the same variables (\(x\) and \(y\)) with identical exponents, so they are like terms. Once identified, you can proceed to add or subtract the coefficients of these terms to streamline the expression.

Coefficients

In algebra, the coefficient is the number that is multiplied by a variable in an algebraic expression. It's an essential concept, as it tells us how 'strong' the variable's presence is within an expression. In our example of \( -9xy + 2xy \), \(-9\) and \(2\) are coefficients of the like terms \(xy\).

To combine these like terms, you simply add (-9) and (2), which are the numeric multipliers of the variable components. Adding the coefficients correctly is crucial to simplifying expressions accurately. So, in our case, \(-9 + 2\) equals \(-7\), leading to the simplified expression \(-7xy\). This process reduces the complexity of the original equation without altering its intrinsic value or truth.

Algebraic Expressions

An 'algebraic expression' consists of constants, variables, and operators combined to convey a computation or to represent a particular relationship. An algebraic expression does not include an equality sign, so it isn't an equation, but it still carries meaningful information. In the expression \( -9xy + 2xy \), the variables are \(x\) and \(y\), both of which are multiplied by their respective coefficients.Variables can have different powers (exponents), and when they have the same base and exponent, they can be combined if the operation (like addition or subtraction) is permissible. Combining like terms within an algebraic expression is a foundational skill in algebra, as it simplifies the expression and prepares it for further operations that might include solving for variable values or manipulating the expression to reveal some particular insight.