Problem 314 - Intermediate Algebra

Question

In the following exercises, simplify. $$ -22 p+17 q+(-35 p)+(-27 q) $$

Step-by-Step Solution

Answer

-57p - 10q

Step 1: Identify Like Terms

First, identify the like terms in the expression. Like terms are terms that have the same variables. In this case, they are \(-22p\) and \(-35p\) as well as \(17q\) and \(-27q\).

Step 2: Combine Like Terms

Combine the like terms by performing the addition and subtraction operations on the coefficients: \(-22p - 35p = -57p\) \(17q - 27q = -10q\)

Step 3: Rewrite the Expression

Replace the original like terms with the simplified terms: \(-22p + 17q + (-35p) + (-27q) = -57p - 10q\)

Key Concepts

Combining Like Terms

When working with algebraic expressions, combining like terms is a vital step. But what does this really mean? Simply put, like terms are terms that have the same variable(s) raised to the same power. For example, the terms \(-22p\) and \(-35p\) are like terms because they both contain the variable \(p\) raised to the same power (which is 1 here).
Similarly, \(17q\) and \(-27q\) are like terms.
Combining these like terms means adding or subtracting their coefficients. This helps in simplifying the expression to a more manageable form.
Example in the exercise:

• The like terms \(-22p -35p\) combine to form \(-57p\)
• The like terms \(17q - 27q\) combine to form \(-10q\)

Identifying Like Terms

Before you can combine like terms, you need to identify them. Like terms have the same variable parts. This means the variables need to match exactly, both in terms of letter and their exponents. For example:

\( 3x^2 \) and \( -5x^2 \) are like terms because they both have \( x^2 \).
But \( 3x \) and \( 3x^2 \) are not like terms because the exponents of the variable \( x \) do not match.
In our given exercise, we identified \(-22p\) and \(-35p\) as like terms, as well as \(17q\) and \(-27q\). This is a crucial step. Without correctly identifying the like terms, you cannot proceed to simplify the expression.

Algebraic Simplification

Once you've combined like terms, you'll find that algebraic simplification becomes straightforward. Simplification is essentially making an expression as simple as possible. This often involves combining like terms, reducing coefficients, and removing any unnecessary parentheses.
In our example, after identifying and combining the like terms, we rewrote the expression from:
\( -22p + 17q + (-35p) + (-27q) \)
to
\( -57p -10q \)
By reducing the expression to \(-57p -10q\), we've simplified it. There's less clutter, and it's often easier to understand or use in further calculations. This makes algebraic simplification a useful tool in both solving equations and in understanding algebraic expressions better.