Problem 21 - Algebra for College Students

Question

Determine whether each is an expression or an equation. Simplify any expressions, and solve any equations. $$ 5(x+3)-8(2 x-6) $$

Step-by-Step Solution

Answer

The simplified expression is -11x + 63.

Step 1: Identify the given mathematical sentence

The provided mathematical sentence is 5(x+3)-8(2x-6). This sentence does not contain an equal sign, so it is an expression.

Step 2: Distribute the constants

Apply the distributive property to remove the parentheses: 5(x+3) - 8(2x-6) = 5x + 15 - 16x + 48.

Step 3: Combine like terms

Combine the terms involving x and the constant terms: 5x - 16x + 15 + 48 = -11x + 63.

Key Concepts

Distributive Property

When simplifying algebraic expressions, the distributive property is a really useful tool. It helps you remove parentheses by distributing a multiplier to each term inside the parentheses. This is how it works: For any numbers a, b, and c, the distributive property states that:
a(b + c) = ab + ac
. You simply multiply the term outside the parentheses by each term inside the parentheses. Let's look at an example:
In the expression 5(x+3)-8(2x-6), you can apply the distributive property twice.
5(x + 3) becomes 5x + 15, and -8(2x - 6) becomes -16x + 48. After using the distributive property, the expression is 5x + 15 - 16x + 48. The next step is to combine like terms!

Combining Like Terms

Combining like terms means adding or subtracting terms that have the same variable. This helps simplify the expression. Like terms are terms that have the same variable and the same exponent. For example,

• 5x and -16x are like terms because they both have the variable x.
• 15 and 48 are like terms because they are both constants (terms without variables).

To combine like terms, just add or subtract their coefficients. In the expression 5x + 15 - 16x + 48, you would combine:

• 5x - 16x which gives -11x and
• 15 + 48 which gives 63.

So, the simplified expression is -11x + 63. Combining like terms makes the expression easier to work with and understand.

Simplifying Expressions

Simplifying an algebraic expression means rewriting it in its simplest form. It often involves using the distributive property and combining like terms. When you simplify, you make the expression shorter and easier to work with.

• First, you apply the distributive property to get rid of parentheses.
• Then, you identify and combine like terms.

In our example, we start with the expression 5(x + 3) - 8(2x - 6):

• After using the distributive property, it becomes 5x + 15 - 16x + 48.
• Then, by combining like terms, we get
  -11x + 63.

The final expression -11x + 63 is simplified. Simplifying expressions helps you solve problems more easily and is a key skill in algebra. Practicing these steps will make you more confident in handling algebraic expressions.