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Chapter 3: Problem 1

Identify the terms, like terms, coefficients, and constants in expression. \(4 x+3+5 x+y\)

Short Answer

Expert verified
Terms: \(4x, 3, 5x, y\). Like terms: \(4x, 5x\). Coefficients: 4, 5, 1. Constant: 3.

Step by step solution

01

Break Down the Expression

First, list all the terms in the given expression. The expression is \(4x + 3 + 5x + y\). The terms are the components separated by the plus sign: \(4x\), \(3\), \(5x\), and \(y\).
02

Identify Like Terms

Like terms have the same variable raised to the same power. Identify like terms from the list of terms. Here, the like terms are \(4x\) and \(5x\) because they both contain the variable \(x\).
03

Identify Coefficients

Coefficients are numbers in front of variables in a term. For \(4x\), the coefficient is \(4\) and for \(5x\), the coefficient is \(5\). For the term \(y\), the coefficient is \(1\) (since it's unwritten, it is implied).
04

Identify Constants

Constants are terms without any variables. In the expression \(4x + 3 + 5x + y\), the constant is \(3\) since it has no variable.

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Key Concepts

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

Like Terms
In algebraic expressions, "like terms" are terms that have the same variables raised to the same power. This means that their variable parts (not the coefficients) look exactly the same. For instance, in the expression \(4x + 3 + 5x + y\), \(4x\) and \(5x\) are like terms because they both have the variable \(x\) raised to the first power. On the other hand, \(y\) does not have any like terms among the given terms because no other term contains the variable \(y\).
  • Terms with the same variable(s) and exponent(s) are like terms.
  • You can combine like terms by adding or subtracting their coefficients.
Combining like terms helps simplify expressions, making calculations easier. Always keep an eye out for these, as they often pop up in equations and polynomial expressions.
Coefficients
Coefficients are the numbers that appear in front of the variables in algebraic terms. They are the "big numbers" or multipliers and tell you how many of each variable you have. In the expression \(4x + 3 + 5x + y\), the coefficients are the numbers directly before the variables:
  • For \(4x\), the coefficient is \(4\).
  • For \(5x\), the coefficient is \(5\).
  • For \(y\), even though it looks like it doesn鈥檛 have a number, the coefficient is actually \(1\) (this is implied if not written).
Recognizing coefficients is crucial because they play a big role in operations like multiplication and addition within algebraic expressions. They help in combining like terms and solving equations.
Constants
Constants in algebraic expressions are the numbers that stand alone, without any attached variable. They do not change because they are not affected by variables. Consider the expression \(4x + 3 + 5x + y\): here, \(3\) is the constant because it doesn鈥檛 have any variable attached to it.
  • Constants remain unaffected by changes in the values of variables.
  • They represent fixed values in expressions and equations.
Understanding constants is key when evaluating or simplifying expressions, as they often simplify calculations by providing a fixed reference point.

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