Problem 19 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 7 y-9 y $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: $$7y - 9y$$. Answer: $$-2y$$.

Step 1: Identify like terms

In the given expression, $$7y - 9y$$, both of the terms are like terms since they both involve the variable y.

Step 2: Combine like terms

To simplify the expression, we will subtract the coefficients of the y terms. So we have: $$7y - 9y = (7-9)y$$

Step 3: Perform the subtraction

Now, we will subtract the coefficients: $$(7-9)y = -2y$$ So the simplified algebraic expression is $$-2y$$.

Key Concepts

Combining Like Terms

When working with algebraic expressions, one essential skill is to combine like terms. Like terms are terms within an expression that have the same variable raised to the same power. For example, in the expression \(7y - 9y\), both terms are like terms because they share the variable \(y\).To combine like terms, you need to look at the variables and their exponents:

• The variables in \(7y\) and \(9y\) are the same (\(y\)) and are both raised to the power of one.
• This means they can be added or subtracted from each other.

Combining like terms simplifies expressions and reduces the number of terms, making the expression easier to understand and work with.

Simplifying Expressions

Simplifying expressions involves rewriting them in their simplest form. This means reducing the expression so it contains the fewest possible terms without changing its value. In our example, we start from \(7y - 9y\). After identifying that both terms are like terms, we proceed to simplify by combining them:

• First, acknowledge the like terms: in our case, \(y\) terms.
• Perform operations to combine them, such as addition or subtraction.

Once you combine the like terms, the expression is already simpler. The aim is always to reach a format that is easy to interpret and utilize in further calculations or for solving equations.

Subtraction of Coefficients

In algebra, coefficients are the numbers located in front of the variables. In our expression \(7y - 9y\), the coefficients are 7 and 9. Subtracting coefficients is critical when simplifying expressions like this one.Here's how it works:

• Identify the coefficients from the like terms: 7 in \(7y\) and 9 in \(9y\).
• Perform subtraction: Calculate \(7 - 9\), which results in \(-2\).
• Since the variable \(y\) is common in both terms, attach it back to the result: \(-2y\).

Subtracting the coefficients directly gives the expression its simplified form, \(-2y\), ensuring the expression is concise yet retains its original mathematical meaning.