Problem 15 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-2-(5-3 p)\)

Step-by-Step Solution

Answer

-7 + 3p

Step 1: Distribute the negative sign

Start by distributing the negative sign across the parentheses. This changes the signs of the terms inside the parentheses: -2 - (5 - 3p) = -2 - 5 + 3p

Step 2: Combine like terms

Next, combine the constant terms -2 and -5: -2 - 5 + 3p = -7 + 3p

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows you to multiply a term outside the parentheses by each term inside the parentheses. This property is expressed like this: a(b + c) = ab + ac In the given problem, you need to distribute the negative sign across the terms inside the parentheses: -2 - (5 - 3p) The negative sign is equivalent to multiplying by -1. So, you distribute -1 to each term inside the parentheses: -1 * 5 = -5
-1 * -3p = +3p Therefore, applying the distributive property, we have: -2 - 5 + 3p

Combining Like Terms

After distributing the negative sign, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. Constant terms (numbers without variables) are also like terms with each other. In our example: -2 - 5 + 3p First, identify the like terms: -2 and -5 are both constants. We combine them by simple addition/subtraction: -2 - 5 = -7 Now, bring down the remaining term that has the variable: -7 + 3p This step simplifies the expression significantly and makes it easier to understand and use in further calculations.

Constant Terms

Constant terms are numbers without any variables attached to them. They are straightforward to work with and only require basic arithmetic operations like addition or subtraction. In the problem, the constant terms were -2 and -5: -2 and -5
These constants were combined to form: -7 When simplifying expressions, it's essential to handle constant terms separately from variable terms to keep your calculations clear and correct. To recap, the fully simplified expression of the original problem is: -7 + 3p