Problem 48 - Elementary Algebra

Question

Simplify. $$ -12 a b-3+4 a b-20 $$

Step-by-Step Solution

Answer

The simplified expression is \(-8ab - 23\).

Step 1: Identify Like Terms

To simplify the expression \(-12ab - 3 + 4ab - 20\), first identify the like terms. Like terms are terms that have the same variable parts. Here, the like terms are \(-12ab\) and \(4ab\). The constants \(-3\) and \(-20\) are also like terms.

Step 2: Combine Like Terms with Variables

Now, combine the like terms with variables \(-12ab\) and \(4ab\) by adding their coefficients. \(-12 + 4 = -8\). So, \(-12ab + 4ab = -8ab\).

Step 3: Combine Constant Terms

Next, combine the constant terms \(-3\) and \(-20\). \(-3 - 20 = -23\).

Step 4: Write the Simplified Expression

Finally, write down the simplified expression combining both results from the previous steps: \(-8ab - 23\).

Key Concepts

Like Terms

When working with algebraic expressions, it's essential to understand the term "like terms." Like terms are components of an expression that have the exact same variables raised to the same powers. This similarity allows us to combine them to simplify the expression more efficiently.

For instance, in the expression \(-12ab - 3 + 4ab - 20\), the terms \(-12ab\) and \(4ab\) are like terms. They both contain the same variables, "a" and "b," and are raised to the same power (which is 1 in this case). Even though their coefficients are different, their variable components are identical, enabling us to group them together during simplification.

Recognizing like terms is crucial for simplifying expressions successfully since it helps in reducing the number of terms in the expression. Remember, terms without the same variable parts cannot be combined, and each term is separated by addition or subtraction in the expression.

Combining Like Terms

Once you've identified the like terms within an expression, the next step is to combine them to simplify the expression further. This essentially means performing arithmetic on the coefficients of the like terms while keeping their variable parts unchanged.

Consider the terms \(-12ab\) and \(4ab\) from our expression \(-12ab - 3 + 4ab - 20\). To combine these, focus on their coefficients, which are -12 and 4, respectively.

• Calculate: \(-12 + 4 = -8\).
• Then, combine the like terms: \(-12ab + 4ab = -8ab\).

This process reduces the expression and brings clarity, making it easier to understand and work with further. Combining like terms streamlines expressions by minimizing layers of complexity and clutter caused by multiple similar terms.

Constants

Constants in an algebraic expression are terms that do not contain any variables. They are fixed values that remain the same regardless of changes in the equation or problem.

In the expression \(-12ab - 3 + 4ab - 20\), \(-3\) and \(-20\) are constants. Just like how we combined like terms that have variables, constants can be combined through simple addition or subtraction.
To handle the constants in this expression:

• Combine the constants: \(-3 - 20 = -23\).

This indicates that, when you simplify an expression, constants can be treated independently from the variable terms, allowing them to be simplified effectively on their own. Ultimately, constants just add a numerical value to the final simplified expression, contributing to the holistic understanding of the problem.