Problem 67 - Elementary Algebra

Question

For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors. $$ 7 y ; y $$

Step-by-Step Solution

Answer

Answer: The coefficient of the group of factors "y" in the term "7y" is 7.

Step 1: Identifying the term

The given term is 7y.

Step 2: Identifying the group of factors

The given group of factors is y.

Step 3: Finding the coefficient

Since 7y can be written as 7 * y, the coefficient of the group of factors y is 7.

Key Concepts

Algebraic Terms

In algebra, an algebraic term is essentially the building block of algebraic expressions. It is composed of numbers, variables (letters), and sometimes exponents that are all multiplied together. An example of an algebraic term is 7y, which is discussed in the textbook problem.

A term like 7y is comprised of two important elements: the coefficient, which in this case is 7, and the variable or group of variables, here represented by y. The coefficient is a numerical factor that multiplies the variable, and it provides valuable information about the term, such as its scale or direction if it represents a vector quantity in physics.

It's also worth noting that a term can be a constant, without any variables, like the number 3, or it can have multiple variables and exponents, such as 2x^2y, where 2 is the coefficient, x and y are variables, and 2 is also the exponent on x.

Factors in Algebra

A factor in algebra refers to a quantity that divides another quantity without leaving a remainder, much like in arithmetic. However, in algebra, factors can include both numbers and variables. For an algebraic term, factors are what you would multiply together to get that term.

For instance, the term 7y has factors 7 and y. When multiplied together, these factors give you the original term. Identifying factors in algebra is crucial when simplifying expressions, solving equations, and factoring polynomials.

Important to Remember

Every algebraic term can be broken down into a product of its factors, and recognition of these factors is a key step in many algebraic methods, such as factorization and the simplification of fractions.

Coefficient Extraction

The process of coefficient extraction is identifying and separating the numerical part of a term from its variables. It's essentially 'extracting' the number that's multiplying the variables within a term.

In the original textbook problem, the term given is 7y and the coefficient we need to extract corresponds to the factor y. Since the term is actually the result of 7 multiplied by y, this makes 7 the coefficient of y.

How to Extract Coefficients

To extract the coefficient, observe the term and identify which factor is a number and which are variables or groups of variables. The coefficient is the standalone number, or in cases where there's a multinomial factor, it could be an algebraic expression itself. Understanding coefficient extraction is essential as it helps in simplifying algebraic expressions and solving algebraic equations effectively.