Problem 39 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ -9(a+7) $$

Step-by-Step Solution

Answer

Answer: The result of the multiplication and simplification is -9a - 63.

Step 1: Distribute the constant

To distribute the constant (-9), we need to multiply it with each term inside the parentheses: $$ -9(a+7) = -9 \cdot a + (-9)\cdot 7 $$

Step 2: Perform the multiplications

Multiply the constant with each term: $$ -9 \cdot a + (-9)\cdot 7 = -9a - 63 $$

Step 3: Combine like terms (if any)

In this case, there are no like terms to combine further. So, the final simplified expression remains: $$ -9a - 63 $$

Key Concepts

Distribution Property

The distribution property, also known as the distributive law, is a fundamental concept in algebra. It allows you to multiply a single term by each term within a parenthesis. This is particularly useful when simplifying expressions or solving equations. The general rule for the distributive property is:

• \(a(b + c) = ab + ac\)

When applying the distributive property, it’s important to remember to multiply every term inside the parentheses by the term outside. In our exercise, we have to distribute \(-9\) across \((a+7)\).
This means:

• First, multiply \(-9\) by \(a\), which gives us \(-9a\).
• Second, multiply \(-9\) by \(7\), resulting in \(-63\).

Through distribution, we have successfully broken down a seemingly complex expression into a simpler form, allowing for easier computation and further simplification.

Combining Like Terms

Combining like terms is the next step in simplifying an expression after using the distribution property. "Like terms" are terms that contain the same variables raised to the same powers. This means you can only combine terms that share the same variable components.

• For example, \(3x\) and \(5x\) are like terms, but \(3x\) and \(5y\) are not.

In our specific problem, after applying distribution \(-9a - 63\), there are no like terms present since there is only one term with a variable \(a\).
Therefore, combining like terms isn't needed, but it's vital to check for them as this step helps in further simplifying the expression when possible.

• Look for terms with the same variable(s) like \(x\), \(y\), etc.
• Add or subtract the coefficients while keeping the common variable attached.

Combining like terms can greatly reduce the complexity of algebraic expressions and prepare them for other operations such as solving for a variable.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. This process often involves both distributing and combining like terms.

• First, employ the distribution property to remove any parentheses.
• Next, combine any like terms you find.

In our exercise, once we applied the distributive property, we reached the expression \(-9a - 63\).
Since no like terms were present to combine, this expression is already in its simplest form.

• Simplifying makes expressions easier to work with, especially in larger equations.
• This process helps reveal underlying structures of algebraic equations, aiding in solving them.

Simplifying expressions is a core skill in algebra that supports solving equations and understanding more complex algebraic concepts.