Problem 76 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$5+9 y-29 y$$

Step-by-Step Solution

Answer

The simplified algebraic expression of \(5+9y-29y\) is \(5 - 20y\).

Step 1: Identify Like Terms

A 'like term' refers to a term that has the same variable raised to the same power. In this expression, '9y' and '-29y' are like terms because both have the same variable y.

Step 2: Combine Like Terms

Using the properties of real numbers, the like terms can be combined by adding or subtracting the coefficients. '9y' minus '29y' gives '-20y'.

Step 3: Rewrite the Expression

Lastly, replace the like terms with the simplified term. Thus, the expression becomes \(5 - 20y\).

Key Concepts

Like Terms

In algebra, understanding 'like terms' is crucial for simplifying expressions. Like terms are terms that contain the same variables raised to the same power. This means that not only should the variables be identical, but their exponents must also match.

For instance, in the expression given: \( 5 + 9y - 29y \), both \(9y\) and \(-29y\) are considered like terms. They share the variable \( y \) raised to the first power. The constant term \( 5 \) does not have a variable attached, so it is not a like term with \(9y\) or \(-29y\).

Recognizing like terms simplifies algebraic operations significantly. It allows us to combine them efficiently, which is a key step in reducing complex expressions.

Combining Like Terms

Once like terms are identified, the next step is to combine them. This involves performing arithmetic operations on their coefficients, while keeping the common variable and its power unchanged.

Let's look at our example: \( 9y - 29y \). To combine these terms:

• Add or subtract their coefficients. Here, you subtract \(29\) from \(9\) which results in \(-20\).
• Retain the shared variable \( y \).

So, \( 9y - 29y = -20y \).

By combining like terms, you streamline the expression into a simpler form, making it easier to handle. Always ensure to perform the operation on the coefficients while ensuring the variable part remains intact.

Real Numbers Properties

To simplify algebraic expressions effectively, one often relies on the properties of real numbers. These mathematical properties help in the systematic combination of numbers and terms.

Some key properties include:

• Commutative Property: This states that numbers can be added or multiplied in any order. For example, \( a + b = b + a \).
• Associative Property: This indicates that the grouping of numbers can be changed without affecting the outcome. For example, \( (a + b) + c = a + (b + c) \).
• Distributive Property: This property allows you to multiply a sum by multiplying each addend separately and then adding the results, such as \( a(b+c) = ab + ac \).

In the context of our expression \( 5 + 9y - 29y \), these properties guide the simplification process by ensuring each step remains valid and logical. For instance, the associative property confirms that regardless of how terms are grouped, the result remains consistent.