Problem 59 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses Then simplify the result. See Example 4 . \(-9(4 x+8)+2\)

Step-by-Step Solution

Answer

The simplified expression is \\(-36x - 70\\).

Step 1: Apply the Distributive Property

The expression \(-9(4x + 8) + 2\) can be expanded using the distributive property. This involves multiplying \-9\ with each term inside the parentheses. Compute as follows: \(-9 imes 4x = -36x\) \(-9 imes 8 = -72\). The expression becomes: \(-36x - 72 + 2\).

Step 2: Combine Like Terms

Now, look for like terms in the expression. The terms \-72\ and \+2\ are constants and can be combined. Compute: \(-72 + 2 = -70\). This results in the simplified expression: \(-36x - 70\).

Key Concepts

Combining Like Terms

In algebra, an important skill is the ability to combine like terms. Like terms are terms that have the same variables raised to the same power. For example, in the expression \(-36x - 72 + 2\), "\(-36x\)" and "\(x\)" are not like terms because they don't match. On the other hand, "\(-72\)" and "\(+2\)" are like terms because they are both constants, meaning they only contain a number with no variable attached. To combine them, simply add or subtract the numerical values:

• \(-72 + 2 = -70\)

Always remember:

• Only add or subtract terms that have identical variable parts.
• Check that the powers are the same for variable terms to treat them as like terms.

When you correctly combine like terms, it makes the expression simpler and easier to work with for further algebraic manipulation.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. This often involves both expanding and combining like terms. Let's look at the expression \(-36x - 72 + 2\). Here, the term "\(-36x\)" stands alone as it doesn't have any like terms. However, "\(-72\)" and "\(+2\)" are combined to produce:

• \(-70\)

The resulting expression is \(-36x - 70\). It's simplified because all like terms have been combined, and there are no unnecessary parentheses or complex operations remaining.
Simplifying is an essential process in solving algebraic problems because:

• It reduces the complexity of expressions.
• Makes subsequent calculations easier and more straightforward.

Think of it like organizing a cluttered room, discarding unnecessary items or combining items to tidy up.

Algebraic Expressions

Algebraic expressions form the foundation of algebra. They consist of numbers, variables, and operations, representing mathematical relationships and computations. For example, the expression \(-9(4x + 8) + 2\) involves variables 'x', constants, and operations. Each component serves a role:

• Constants: Numbers like \(-9\), \(+8\), and \(+2\) that do not change.
• Variables: Represented by symbols like \(x\), they stand in for unknown values that can vary.
• Operations: Addition, subtraction, and multiplication are used to form and manipulate expressions.

Understanding algebraic expressions involves recognizing these components and how they interact, allowing you to perform operations like expansion using the distributive property and simplification. Recognizing the distinct parts helps clarify complex problems, making it easier to manage and solve them through algebraic processes.