Problem 30 - Elementary and Intermediate Algebra

Question

To prepare for Section 9.3, review simplifying expressions \((\text { Section } 1.8)\) Simplify. [ 1.8] $$ 3(2 a+4 b)+(5 a-12 b) $$

Step-by-Step Solution

Answer

8a - 20 b

Step 1: Distribute the 3 inside the parentheses

First, we need to distribute the coefficient 3 to the terms inside the parentheses:

Step 2: Combine like terms

Next, we combine the like terms:

Step 3: Simplify the expression

Finally, we simplify the combined expression to get a single simplified expression:

Key Concepts

distributive property

The distributive property is a fundamental concept in algebra that allows you to break down expressions for easier simplification. It states that: \[ a(b + c) = ab + ac \] This means you multiply the outside term by each of the terms inside the parentheses.
In our exercise, we have: \[ 3(2a + 4b) \] Applying the distributive property, we get: \[ 3 \times 2a + 3 \times 4b \] which simplifies to: \[ 6a + 12b \] This breaks a complex expression into more manageable parts, making it easier to combine and simplify further.
Mastering the distributive property is key to progressing in algebra and solving more complex equations.

combining like terms

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that have identical variable parts.
For example, in our exercise: \[ 6a + 12b + 5a - 12b \] Notice we have two sets of like terms: \[ 6a \text{ and } 5a \] are both terms with 'a', and \[ 12b \text{ and } -12b \] are both terms with 'b'.
To combine like terms, add or subtract the coefficients of these terms. For the 'a' terms, we have: \[ 6a + 5a = 11a \] For the 'b' terms, we have: \[ 12b - 12b = 0 \] The resulting simplified expression is: \[ 11a \] This method reduces the complexity of the expression, making it simpler and easier to evaluate.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication). They form the building blocks of algebra.
In our exercise, we started with: \[ 3(2a + 4b) + (5a - 12b) \] This is an algebraic expression because it includes variables (a and b) and constants (the numbers) connected by operation symbols.
When simplifying algebraic expressions, your goals are:

• Apply properties like the distributive property
• Combine like terms
• Simplify to the fewest terms possible

Doing so transforms a complex algebraic expression into a simpler form.
Understanding and manipulating algebraic expressions is pivotal in solving equations and problems in algebra. This foundation will help you tackle more advanced math topics confidently.