Problem 63 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$7(3 a+2 b)+5(4 a+2 b)$$

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression is \(41a + 24b\).

Step 1: Distribute the constants

Multiply each term inside both parentheses by the corresponding constants in front of the parentheses. It should look like this: \(21a + 14b + 20a + 10b\)

Step 2: Combine like terms

Now that we've distributed the constants, look for like terms and combine them. 'Like terms' refers to those terms that have the same variable and the same exponents. In this case, those terms are \(21a\) and \(20a\), and \(14b\) and \(10b\). After combining these we get \(41a + 24b\)

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that helps to simplify expressions. It allows us to remove parentheses by distributing, or spreading, a multiplication across terms that are added or subtracted inside the parentheses. Here's how it works:

• Given an expression like \(a(b + c)\), the Distributive Property lets us multiply \(a\) by both \(b\) and \(c\).
• So \(a(b + c)\) becomes \(ab + ac\).
• This step ensures all parts of the expression are accounted for in the multiplication.

Applying this property helps in transforming complex expressions into simpler ones, making them easier to handle and solve. In the example \(7(3a+2b)+5(4a+2b)\), we use the Distributive Property as follows:

• First, multiply 7 by each term inside the first parenthesis, resulting in \(21a + 14b\).
• Then, do the same with 5 in the second parenthesis to get \(20a + 10b\).

This step is crucial as it prepares the terms so they can be combined effectively in the next step.

Combining Like Terms

Combining Like Terms is another essential technique in algebra for simplifying expressions. Once you've distributed the terms in an expression, the next step is to "combine" or add together those terms that have identical variable parts. But what exactly are "like terms"?

• Like terms have the same variables raised to the same powers. For example, \(3x\) and \(5x\) are like terms while \(3x\) and \(3y\) are not because they have different variables.
• Combining them simply means adding or subtracting their coefficients.

Let's look at our example from above:

• After distributing, we have \(21a + 14b + 20a + 10b\).
• You can see that \(21a\) and \(20a\) are like terms, as are \(14b\) and \(10b\).
• Add the coefficients of \(a\) together to get \(41a\), and do the same for \(b\), resulting in \(24b\).

Combining like terms helps streamline expressions and is a vital step before simplifying further or solving equations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation signs. Unlike equations, which show equality, algebraic expressions do not include an equal sign. Expressions are foundational in algebra and used in various ways, such as in equations or functions.

• Expressions can be as simple as \(3x\) or as complex as \(7(3a + 2b) + 5(4a + 2b)\).
• They consist of terms that are parts of the expression separated by addition or subtraction.
• Each term is made up of constants, coefficients, and variables.

Working with algebraic expressions often involves simplifying them, which we do by applying the Distributive Property and combining like terms. Simplifying expressions is critical because it makes them easier to work with, particularly in equations where solving for a specific variable is the goal.
Understanding the structure and components of algebraic expressions is essential for moving on to more advanced algebra topics where you'll be performing even more operations with them.