Problem 52 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 7(a+2)+4 $$

Step-by-Step Solution

Answer

Question: Simplify the expression $7(a + 2) + 4$. Answer: The simplified expression is $7a + 18$.

Step 1: Distribute the factor

To distribute the factor of 7, we need to multiply it with each term inside the parentheses, so we get: $$ 7 \times a + 7 \times 2 + 4 $$

Step 2: Simplify the expression

Now, we'll simplify the expression by multiplying the numbers and combining the constant terms, which are the constants without a variable (7 * 2 and 4) and we have: $$ 7a + 14 + 4 $$

Step 3: Combine the constant terms

Finally, combine the constant terms 14 and 4 by adding them: $$ 7a + (14+4) $$ This results in the simplified expression: $$ 7a + 18 $$

Key Concepts

Distributive Property

The distributive property is like having a secret weapon to unlock all sorts of expressions. It allows us to multiply a single term by each term inside a parenthesis. Think of it as "distributing" that outside term into everything within. In mathematics, it can be expressed as:

• \( a(b + c) = ab + ac \)

For example, let's use the distributive property in the expression given in the original exercise: \[ 7(a+2) + 4 \].
The term \( 7 \) is the distributor. You'll distribute the \( 7 \) to both \( a \) and \( 2 \) separately. This means you'll perform the operations: - \( 7 \times a \)- \( 7 \times 2 \)
By using the distributive property, the expression becomes: \[ 7a + 14 + 4 \]. This step sets you up perfectly to simplify the rest. The beauty of the distributive property is that it helps to break down more complicated expressions into manageable bits.

Simplifying Expressions

Simplifying expressions is about making things simpler and more readable. When an expression contains various terms and operations, simplification helps us combine like terms.

In the exercise, after using the distributive property, we got the expression: \[ 7a + 14 + 4 \].
Here, "simplifying" means we need to combine any like terms, reduce terms involving numbers, and generally make the expression as clean as possible.

• The term \( 7a \) can't be simplified further since it has a variable which isn't shared with other terms.
• The constant terms \( 14 \) and \( 4 \) can be added together because they don't have variables attached.

When you add these constants, the expression becomes:\[ 7a + 18 \]. This is a much simpler version of what we started with. When simplifying, always look for like terms (terms with the same variable) and remember to combine constant terms wherever possible.

Constants in Algebra

In algebra, constants are the numbers standing on their own without knights in shining armor—variables. They often represent specific, fixed values that do not change. In dealing with expressions, identifying constants is important because they can usually be combined or simplified.

Looking at the expression from the exercise: \[ 7a + 14 + 4 \],
\( 14 \) and \( 4 \) are the constants. They do not change based on the variable \( a \).

• Because they are constant, they can simply be added together: \( 14 + 4 = 18 \).
• This ability to combine constants simplifies the overall algebraic expression.

By recognizing that constants can be combined, you're able to clean up your work and make solving algebra problems more straightforward. Constants are often the easiest part to handle in expressions, so they're your friends when simplifying!