Problem 85 - Algebra A Combined Function

Question

Simplify each expression. See Section \(1.8 .\) \(6(2 z+4)+20\)

Step-by-Step Solution

Answer

The simplified expression is \(12z + 44\).

Step 1: Distribute the 6

First, distribute the 6 to both terms inside the parenthesis. This means multiplying 6 by each term inside: \[ 6(2z + 4) = 6 \times 2z + 6 \times 4 \] This simplifies to:\[ 12z + 24 \]

Step 2: Combine Like Terms

Now add the 20 to the expression obtained from Step 1. Since there are no like terms with \(12z\), combine the constants:\[ 12z + 24 + 20 \]Simplify by adding the constants:\[ 12z + 44 \]

Key Concepts

Distributive Property

The distributive property is a key foundational principle in algebra. Learning how to use it effectively can greatly simplify complex expressions.

In the original exercise, we see an example of using the distributive property with the expression \(6(2z + 4)\). The key step in applying the distributive property is to multiply the term outside the parenthesis with each term inside.
- Multiply 6 by each term inside the parenthesis: - 6 multiplied by \(2z\) results in \(12z\). - 6 multiplied by 4 gives 24.

By distributing, the expression transforms to \(12z + 24\). Understanding this concept is essential as it helps in dealing with parentheses and converting expressions into a simpler linear format.

Combining Like Terms

Combining like terms is another crucial concept in algebra used to simplify expressions further. Like terms are terms who have the exact same variable raised to the same power, which means they can be combined by simple addition or subtraction.

In the equation from the exercise, after applying the distributive property, we have \(12z + 24\). The next step involves simplifying this expression further by combining like terms.

It's important to note that:- Like terms can only be those with the same variable and exponent, or constants.- Since there are no other terms containing \(z\), \(12z\) remains as it is.
- Only numbers, 24 and 20, can be combined because they are constants. - Adding these constants gives \(44\).

The expression then simplifies to \(12z + 44\). Mastering this skill makes working with any algebraic expressions much easier.

Linear Expressions

Linear expressions are algebraic expressions where the variable(s) is raised to the power of 1. They are termed 'linear' because they graph as a straight line.

The simplified expression from the exercise, \(12z + 44\), is an example of a linear expression. You can recognize these expressions by:- Checking that all the variables, like \(z\) in this case, have an exponent of 1.- Seeing that there are no products of variables such as \(z^2\) or \(xy\).

Understanding linear expressions is important, as they form the foundation for solving more complex equations and inequalities. Working with them involves basic arithmetic operations and requires familiarity with concepts like the distributive property and combining like terms.