Problem 76 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$-7 a+2 a b-7 a+12 a b$$

Step-by-Step Solution

Answer

The simplified expression is \(-14a + 14ab\).

Step 1: Identify Like Terms

To simplify the expression, first identify the like terms. Like terms are terms that have the same variable raised to the same power. Here, the terms \(-7a\) and \(-7a\) are like terms, and \(2ab\) and \(12ab\) are like terms.

Step 2: Combine Like Terms for 'a'

Combine the like terms \(-7a\) and \(-7a\). Since they have the same coefficient and variable, you can add them: \(-7a + (-7a) = -14a\).

Step 3: Combine Like Terms for 'ab'

Next, combine the like terms \(2ab\) and \(12ab\). Add the coefficients of these terms together: \(2ab + 12ab = 14ab\).

Step 4: Write the Simplified Expression

After combining the like terms, the expression simplifies to \(-14a + 14ab\).

Key Concepts

Like Terms

In algebraic expressions, identifying like terms is an essential skill that simplifies calculations. Like terms are terms that contain the same variable parts. It is important to note that these variables must be raised to identical powers, though the coefficients in front of them can vary. For example:

• In the expression \(-7a + 2ab - 7a + 12ab\), the terms \(-7a\) and \(-7a\) are like terms because they both contain the variable \(a\) to the first power.
• Similarly, the terms \(2ab\) and \(12ab\) are like terms because they both contain the variables \(a\) and \(b\) to the first power.

Recognizing like terms helps to simplify expressions, a crucial step before solving equations. Always remember, only terms with identical variable parts qualify as like terms.

Combining Like Terms

Combining like terms is the process of adding or subtracting the coefficients of like terms to simplify an expression. This is done after identifying the like terms. It involves three straightforward steps:

1. **Select Like Terms:** Identify which terms in the expression are "like" each other, which we discussed earlier.
2. **Add or Subtract Coefficients:** Take their coefficients and either add or subtract them, as indicated by the operation in the expression.
3. **Rewrite the Expression:** Once combined, write the expression in its simplified form.

As in the example provided:

• For the repeated terms \(-7a\), combine them to get \(-14a\) by adding: \(-7a + (-7a) = -14a\).
• Similarly, for the terms \(2ab\) and \(12ab\), add their coefficients: \(2ab + 12ab = 14ab\).

The end result of combining these gives you a simplified algebraic expression: \(-14a + 14ab\). This practice simplifies the task of solving equations by reducing the number of terms.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations (like addition and subtraction) combined together. These expressions can represent complex mathematical ideas in a more simplified symbolic form, key for solving problems across different areas of algebra. For instance, take the expression in the example: \(-7a + 2ab - 7a + 12ab\).

Key components of algebraic expressions include variables - which are symbols representing unknown or variable quantities; coefficients - which are numbers placed in front of variables indicating their magnitude; and constants - numbers without attached variables.

• **Variable Part:** In this expression, \(a\) and \(ab\) are the variable parts.
• **Coefficients:** Numbers such as \(-7\), \(2\), and \(12\) act as coefficients.

Learning how to simplify these expressions like in the given task by combining like terms helps understand relationships between quantities. It also paves the way for solving equations, allowing students to predict and explain phenomena both in mathematics and in real-world applications.