Problem 53 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$5 y+3+6 y$$

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression \(5y+3+6y\) is \(11y+3\).

Step 1 - Identify Like Terms

Start by identifying like terms in the expression. 'Like terms' are terms that contain the same variables raised to the same power. In the given algebraic expression \(5y+3+6y\), the like terms are \(5y\) and \(6y\).

Step 2 - Combine Like Terms

Once the like terms have been identified, add these like terms. Add coefficients \((5+6)\) of \(y\) to get \(11y\). The resultant term for this addition is \(11y\).

Step 3 - Rewrite with Simplified Expressions

Since the number 3 does not have any like terms, it will remain as it is. Therefore, replacing the original expression with the simplified version will give final expression \(11y+3\)

Key Concepts

Like Terms

In algebraic expressions, like terms are crucial for simplifying your work. "Like terms" are terms that have the same variable raised to the same power. This means they can be added or subtracted easily because they represent the same quantity.When you look at an expression like \(5y + 3 + 6y\), you should identify "like terms" first. Here, both \(5y\) and \(6y\) are like terms because they share the variable \(y\). On the other hand, the number \(3\) does not share the variable \(y\), and hence, it stands alone. Finding like terms is your first step toward simplification. Keep an eye out for them! They make the process much easier and manageable. This is how you unlock the puzzle of algebraic expressions.

Coefficients

Coefficients are the numbers that stand next to the variables in an algebraic expression. They tell us how many of a particular variable we have. Take \(5y + 3 + 6y\) as an example. Here, \(5\) and \(6\) are coefficients of \(y\). They're like little helpers that represent the quantities of variables. Simply put, they multiply the variable they are attached to.When combining like terms, adding coefficients is the way to go. In our expression, we see \(5y + 6y\). You simply add the coefficients 5 and 6 together. This will give you \(11y\), because

• \(5y\) means five \(y\)'s.
• \(6y\) means six \(y\)'s.
• Adding them gives you eleven \(y\)'s or \(11y\).

Coefficients make it easy to calculate values within expressions.

Simplifying Expressions

Simplifying expressions might sound complex, but it's all about tidying things up. It's like putting your math in order! When you simplify an expression, you make it as straightforward as possible.Starting with \(5y + 3 + 6y\), the goal of simplification is to end with the cleanest form of this expression. This involves:

• Finding like terms: In our case, \(5y\) and \(6y\) were identified.
• Combining the coefficients: Add both coefficients \(5\) and \(6\) to get \(11y\).
• Rewriting the expression with the simplified terms: You rewrite \(5y + 6y + 3\) to \(11y + 3\), since \(3\) has no like terms to combine with.

Simplifying expressions this way ensures your algebra is both elegant and accurate. It makes further calculations easier and less prone to errors.