Problem 2 - Elementary Algebra

Question

Simplify each of the following expressions. $$ 4 y+7 y $$

Step-by-Step Solution

Answer

Answer: $$11y$$

Step 1: Identify like terms

In the given expression, both terms, $$4y$$ and $$7y$$, have the same variable ($$y$$). Hence, they are like terms.

Step 2: Add the coefficients of the like terms

Since both terms are like terms, we can add their coefficients (numbers in front of the variable). $$4y + 7y$$ The coefficients here are 4 and 7. Add the coefficients: $$4 + 7 = 11$$

Step 3: Write the simplified expression

Now that the coefficients have been added, write the final simplified expression with the sum of the coefficients (11) multiplied by the variable ($$y$$): $$11y$$ So, the simplified expression is $$11y$$.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining the elements that look the same, often called "like terms," to make the expression easier to understand and use in calculations. When you simplify an expression like \(4y + 7y\), your goal is to decrease the number of terms by adding or subtracting them when possible.

This process is important because it allows you to handle algebraic equations more efficiently, leading to quicker and more accurate solutions. Here's what you generally do:

• Identify like terms: These are terms that have exactly the same variable raised to the same power.
• Add or subtract the coefficients of these like terms.
• Rewrite the expression using the new coefficient in front of the shared variable.

In our example \(4y + 7y\), both terms contain the variable \(y\), making them like terms. Therefore, you can add their coefficients 4 and 7 to get a single term \(11y\). This simplified expression \(11y\) is much easier to work with in larger equations or real-world problems.

Coefficients

Coefficients are numerical factors that multiply a variable in an algebraic expression. They tell you how many times to count the variable involved. In an expression like \(4y + 7y\), the numbers 4 and 7 are the coefficients.

Understanding coefficients plays a crucial role in combining like terms and simplifying expressions. Here's how you deal with them:

• Identify the coefficients: Look for the numbers in front of the variables; these are your coefficients.
• Combine coefficients of like terms: When the terms are similar (having the same variables), you can add or subtract the coefficients.

For instance, simplifying \(4y + 7y\), you add the coefficients 4 and 7 to get 11. This results in the expression \(11y\). Coefficients are also vital in representing real-world quantities. If, for example, \(y\) represents the number of apples, then \(4y\) means four groups of apples, and \(7y\) means seven groups. Combined, these make \(11y\) or eleven groups in total.

Variables

Variables are symbols, often letters like \(x\) or \(y\), used to represent unknown or generic numbers in an expression or equation. They are fundamental in algebra, allowing us to express and solve various problems flexibly. For instance, in an expression like \(4y + 7y\), \(y\) is the variable.

Understanding the role of variables is essential when working with algebraic expressions:

• Variables can hold different values depending on the context or specific situation.
• They allow for the creation of equations that model real-world situations.

In the example expression \(4y + 7y\), since \(y\) is the same in both terms, it signifies the use of the same quantity or concept across the expression. This consistency is what allows the simplification to happen by combining the terms into \(11y\). Seeing variables as placeholders can help imagine how changing \(y\)'s value might affect the entire expression.