Problem 27 - Elementary Algebra

Question

Solve. $$ 12 x-5+9 x-44 $$

Step-by-Step Solution

Answer

The expression simplifies to \(21x - 49\).

Step 1: Identify Like Terms

The expression given is \(12x - 5 + 9x - 44\). Our first task is to collect like terms. The terms \(12x\) and \(9x\) are like terms as they both contain the variable \(x\). The constants \(-5\) and \(-44\) are like terms as well.

Step 2: Combine the \(x\) Terms

Add the coefficients of the like terms involving \(x\). \(12x + 9x = (12 + 9)x = 21x. \)The expression now becomes: \(21x - 5 - 44\).

Step 3: Combine the Constant Terms

Add the constant terms together. \(-5 - 44 = -49. \)The expression simplifies to: \(21x - 49\).

Step 4: Simplify the Expression

After combining all like terms, the simplest form of the expression is \(21x - 49\).

Key Concepts

Understanding Like Terms

When working with algebraic expressions, it's vital to identify like terms. But what exactly are like terms? Simply put, like terms are terms in an algebraic expression that have exactly the same variable(s) raised to the same power.
For instance, in the expression given, which is \(12x - 5 + 9x - 44\), we notice:

• The terms \(12x\) and \(9x\) both contain the variable \(x\), which makes them like terms.
• The terms without variables, such as \(-5\) and \(-44\), are also like terms because they are both constants.

Recognizing like terms sets the stage for simplification, allowing you to perform operations like addition and subtraction within the expression.
The next steps involve manipulating these terms to simplify the expression into its simplest form.

Combining Like Terms

Once you have identified like terms, the next goal is to combine them. This process simplifies the expression by reducing it to fewer terms. Let's break it down using our example expression, \(12x - 5 + 9x - 44\).

• First, focus on the variable terms \(12x\) and \(9x\).
• To combine them, you add their coefficients: \(12 + 9 = 21\).
• This gives the new term \(21x\).

Now, move to the constant terms, \(-5\) and \(-44\):

• Combine them by adding: \(-5 + (-44) = -49\).

After this combining process, the expression looks much simpler, as it is now \(21x - 49\). Combining like terms is crucial for keeping algebraic expressions manageable and is a fundamental skill in algebra.

Simplification of Expressions

Simplification is all about tidying up an algebraic expression to its most efficient form where no further operations can be performed. After combining all the like terms in our example, the expression \(12x - 5 + 9x - 44\) has been simplified to \(21x - 49\).

Simplification involves:

• Combining terms with the same variables to make single, more manageable terms.
• Reducing constants by performing arithmetic operations to combine them.

The result is a cleaner, more compact expression that is equivalent to the original but easier to analyze and use in calculations.
Remember, a simplified expression should have as few terms as possible, with all like terms combined, making for a more straightforward problem-solving process. Simplification is an essential step in algebra that helps in both clarity and efficiency.