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Chapter 1: Problem 45

Simplify each algebraic expression by combining similar terms. $$-2 x y+12+8 x y-16$$

Short Answer

Expert verified
The simplified expression is \(6xy - 4\).

Step by step solution

01

Identify Like Terms

In the expression \(-2xy + 12 + 8xy - 16\), the like terms need to be identified. Like terms are terms that have the exact same variables raised to the same powers. Here, \(-2xy\) and \(8xy\) are like terms because they both involve \(xy\), and \(12\) and \(-16\) are like terms because they are constant terms.
02

Combine Like Terms

Now that we have identified the like terms, we can combine them. Combine \(-2xy\) and \(8xy\):\[-2xy + 8xy = 6xy\]Combine \(12\) and \(-16\):\[12 - 16 = -4\]
03

Simplified Expression

After combining the like terms, the expression is simplified to:\[6xy - 4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Terms
In algebra, like terms are essential for simplifying expressions. Like terms are terms that contain the same variables, each raised to the same power. For example, in the expression \( -2xy + 8xy + 12 - 16 \, \, -2xy \) and \( 8xy \) are like terms because they both contain the variables \( x \) and \( y \) together.
Think of it like grouping apples with apples and oranges with oranges. Only terms with identical variables and exponents are considered like terms, plain numbers or constants also fit this category. Here, \( 12 \) and \( -16 \) are like terms because they are constants, having no variable attached.
  • Like terms: \( -2xy \, \, 8xy \)
  • Constant terms: \( 12 \, \, -16 \)
Simplifying Expressions
Simplifying algebraic expressions often involves combining like terms into a more condensed form. This process not only makes expressions easier to read but also simpler to work with in calculations.
In our original exercise, simplifying the expression \( -2xy + 12 + 8xy - 16 \) involves combining the like terms that we identified. First, observe the coefficients of the like terms. The coefficients here are these numerical values in front of the terms.
  • For \( xy \): Combine \( -2xy + 8xy \) which results in \( 6xy \).
  • For the constants: \( 12 - 16 \) results in \( -4 \).
After combining, the expression becomes \( 6xy - 4 \). Notice how this expression is much simpler but mathematically equivalent to the original one. This process represents a fundamental skill in algebra.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They can be as simple as a single number or variable, or more complex involving multiple terms.
Understanding how to manipulate these expressions is crucial in algebra. Let's break down the parts:
  • Terms: Each part of an algebraic expression separated by a plus \( + \) or minus \( - \) sign is called a term.
  • Variables: Symbols used to represent numbers we don't know yet, like \( x \) and \( y \).
  • Coefficients: Numbers multiplying the variables, for example, \( -2 \) is the coefficient of \( xy \) in \( -2xy \).
By mastering these components, you can handle any algebraic expression with ease. The goal is to recognize patterns within the expressions to simplify and solve them more efficiently. So, practicing how to simplify expressions will establish a foundation for all future algebraic problem-solving.

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Most popular questions from this chapter

Find the value of \(\frac{h\left(b_{1}+b_{2}\right)}{2}\) for each set of values for the variables \(h, b_{1}\), and \(b_{2}\). (Subscripts are used to indicate that \(b_{1}\) and \(b_{2}\) are different variables.) \(h=9, b_{1}=12\), and \(b_{2}=16\)

State the property that justifies each statement. For example, \(3+(-4)=(-4)+3\) because of the commutative property for addition. $$5(9+(-4))=5(9)+5(-4)$$

Simplify each algebraic expression by combining similar terms. $$-3(x+2)-4(x-10)$$

Simplify each algebraic expression and then evaluate the resulting expression for the given values of the variables. \(5 x-7 y-9 x-3 y\) for \(x=-1\) and \(y=-4\)

Insert parentheses so that \(36+12 \div 3+3+6 \cdot 2\) is equal to \(55 .\)
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