Problem 82 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. \(15-(-3 x)+8 x-(-10)\)

Step-by-Step Solution

Answer

The simplified form of the expression \(15-(-3 x)+8 x-(-10)\) is \(25 + 11x\)

Step 1: Identify the terms in the expression

The terms in the expression are 15, -(-3x), 8x and -(-10).

Step 2: Simplify Subtraction of Negative Numbers

Apply the rule of subtraction of a negative number to the terms -(-3x) and -(-10). This rule states that a minus (-) sign before parentheses that include a negative number is equivalent to addition. Therefore, the expression now becomes: \(15 + 3x + 8x + 10\)

Step 3: Combine Like Terms

Combine the numerical terms and the terms that contain the variable x. Therefore, the numerical terms 15 and 10 combine to form 25, and the terms 3x and 8x combine to form 11x, resulting in the simplified expression: \(25 + 11x\)

Key Concepts

Combining Like Terms

Combining like terms is a fundamental concept in algebra, which involves simplifying expressions by merging terms that have the same variables. Think of it like sorting apples with apples and oranges with oranges. In algebra, similar kinds of terms can be combined to create a simpler expression.
Consider an expression like this: \(3x + 7 + 5x + 2\). Here, you see two terms with the variable \(x\): \(3x\) and \(5x\). Both these terms have the variable \(x\), so they can be combined.
Here's how:

• Identify terms with the same variable: \(3x\) and \(5x\).
• Add the coefficients (numbers in front): \(3 + 5 = 8\).
• The combined form is \(8x\).

Similarly, constant numbers like \(7\) and \(2\) can be combined. Therefore, the expression simplifies to \(8x + 9\). This process helps in making expressions shorter and easier to understand.

Subtracting Negative Numbers

Subtracting a negative number can sometimes be tricky, but it’s crucial for solving and simplifying algebraic expressions. A double negative turns into a positive number. This might sound confusing at first, but it's a powerful tool in algebra.
Imagine you’ve written \(-(-5)\). In this case, subtracting \(-5\) is the same as adding \(5\). The rule is simple: two negatives make a positive.
Let's see how this applies practically:

• In \(15 - (-3x)\), instead of subtracting, you add: \(15 + 3x\).
• For \(-(-10)\), it equals \(+10\).

Learning how to deal with negative numbers efficiently can drastically simplify your process of solving algebraic problems. Remember this rule, and you'll find simplifying expressions much easier!

Algebraic Expressions

Algebraic expressions are a key part of algebra that involve numbers, variables, and operations. They are like sentences in mathematics, telling you the story of a problem.
An algebraic expression might look like \(2x + 5\). Here, \(x\) is the variable representing an unknown number, while \(2\) and \(5\) are constants. The operations like addition or subtraction dictate how to combine these parts.
When simplifying algebraic expressions, it involves a few steps:

• Identify all parts of the expression, like variables and numbers.
• Follow the order of operations: parentheses, exponents, multiplication, division, addition, subtraction (PEMDAS).
• Combine like terms to make it as simple as possible.

In essence, algebraic expressions provide a way to describe mathematical ideas clearly and concisely. By mastering them, you lay the groundwork for tackling more complex algebra and mathematics.