Problem 81 - Intermediate Algebra

Question

Simplify. See Example \(6 .\) $$-3(p-2)+2(p+3)-5(p-1)$$

Step-by-Step Solution

Answer

The simplified expression is \(-6p + 17\).

Step 1: Distribute the Constants

For each term in the given expression, distribute the constant across the terms inside the parentheses. Begin with \(-3(p-2)\), which becomes \(-3p + 6\).Next, distribute through the \(2(p+3)\) which becomes \(2p + 6\).Finally, distribute through the \(-5(p-1)\) which becomes \(-5p + 5\).The expression now becomes:\(-3p + 6 + 2p + 6 - 5p + 5\).

Step 2: Combine Like Terms

Now, combine the like terms starting with the terms that involve \(p\): \(-3p + 2p - 5p = -6p\).Next, combine the constant terms: \(6 + 6 + 5 = 17\).So, the simplified form of the expression is:\(-6p + 17\).

Key Concepts

Combining Like Terms

When working with algebraic expressions, combining like terms is an essential step to simplify an expression. Like terms are terms in an expression that have identical variable parts. For example, in the terms \(-3p\), \(2p\), and \(-5p\), the variable part is \(p\), making them like terms. Properly combining like terms requires adding or subtracting the coefficients while keeping the variable or variables. In the given exercise, after applying the distributive property, we see:

• Together, \(-3p + 2p - 5p\) yield \(-6p\).
• Similarly, the numbers without variables, like the constants 6, 6, and 5, are also like terms, resulting in 17 when combined.

Remember, only terms with the same exact variable component can be combined. Trying to add terms that have different variable parts, such as \(p\) and \(p^2\), will lead to errors. That's a crucial understanding in algebra.

Simplifying Expressions

Simplifying an expression makes it easier to understand and work with. This involves reducing the expression to its simplest form by performing operations and combining like terms. To simplify the original expression \(-3(p-2) + 2(p+3) - 5(p-1)\):

• Start by distributing the numbers outside the parentheses to the terms inside. Distribute the \(-3\) to \(p\) and \(-2\), the \(2\) to \(p\) and \(3\), and the \(-5\) to \(p\) and \(-1\).
• This results in the expression \(-3p + 6 + 2p + 6 - 5p + 5\). Focus on grouping and combining like terms next.

The process of simplifying ensures that the expression is ready for more advanced operations, such as solving equations or further manipulations. A simplified form is neat, compact, and easier to interpret.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. It represents a quantity and allows us to work flexibly with numbers and variables. The exercise begins with an algebraic expression \(-3(p-2) + 2(p+3) - 5(p-1)\). This expression includes:

• The numbers -3, 2, and -5, which are coefficients for the terms inside the parentheses.
• The variables, in this case, \(p\), which acts as placeholders for possible numbers.
• The structure and setup of the expression highlight the relationship between these coefficients and variables.

Understanding algebraic expressions is foundational to algebra. They allow us to describe relationships between quantities and to perform algebraic operations to uncover further information or simplify problems.