Problem 82 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 4 b+9-9 b+9 $$

Step-by-Step Solution

Answer

The simplified expression is \(-5b + 18\).

Step 1: Identify like terms

In the expression \(4b + 9 - 9b + 9\), identify the like terms. 'Like terms' are terms that have the same variable and exponent. In this case, \(4b\) and \(-9b\) are like terms because they both include the variable \(b\). The constants \(9\) and \(9\) are also like terms since they are both constant numbers.

Step 2: Combine the like terms with 'b'

Add the coefficients of the like terms with the 'b' variable. These terms are \(4b\) and \(-9b\). To combine them, you do the arithmetic on their coefficients: \(4 - 9 = -5\). Therefore, combining \(4b\) and \(-9b\) gives \(-5b\).

Step 3: Combine the constant terms

Next, combine the constant terms \(9\) and \(9\). Add these numbers together: \(9 + 9 = 18\). This gives the combined constant term \(18\).

Step 4: Write the simplified expression

Now, combine the results from the previous steps to get the final simplified expression. After combining the like terms, \(-5b + 18\) is the simplified form of the original expression \(4b + 9 - 9b + 9\).

Key Concepts

Simplifying Expressions

When faced with a mathematical expression, it might seem a bit overwhelming at first. But don't worry.
Simplifying expressions simply involves making things neater and easier to understand.
We do this by combining like terms to make the expression as short as possible. To begin the simplification process:

• Look for terms that contain the same variables raised to the same powers. These are your like terms
• Group them together and perform the necessary arithmetic operations

For instance, in the expression \(4b + 9 - 9b + 9\), we've grouped the terms \(4b\) and \(-9b\) separately, and the constants \(9\) and \(9\) separately.
Once you've grouped them appropriately, the expression becomes much simpler to work with and understand.

Identifying Like Terms

The key step in simplifying an expression is to identify like terms. These are terms that have the same variable component.
Even if their coefficients differ, as long as they have the same variable raised to the same power, they can be combined. In our given example:

• \(4b\) and \(-9b\) both have the variable \(b\). This makes them like terms.
• The constant terms, \(9\) and \(9\), are like terms because they have no variables.

It is crucial to grasp this concept because identifying like terms correctly sets the foundation for the rest of the simplification process. Once like terms are identified, you can combine them by adding or subtracting their coefficients.
This will help reduce the clutter and make your algebraic expressions clearer.

Algebraic Expressions

Algebraic expressions are basic building blocks of algebra. They consist of numbers, variables, and arithmetic operators.
Variables are symbols like \(x\), \(y\), or \(b\) that represent numbers. Meanwhile, constants such as \(9\) in our example are fixed values.Writing and understanding algebraic expressions is essential:

• An expression could simply be a number or a variable.
• It might also be a combination, like \(4b + 9\) or \(-9b + 9\).

Understanding the makeup of algebraic expressions enables you to see the relationships between the elements.
Once you're comfortable with this, you'll find it much easier to make sense of more complex problems later on. Learning to simplify these expressions by combining like terms not only reduces complexity but also aids in better problem solving.