Problem 9 - Elementary Algebra

Question

Simplify each of the following expressions by using the distributive property and combining like terms. $$ 5(a+2)+6 a-7+(8+4) \quad(a+3 a+2) $$

Step-by-Step Solution

Answer

Question: Simplify the following expressions using the distributive property and combining like terms. 1. \(5(a+2)+6a-7+(8+4)\) 2. \((a+3a+2)\) Answer: 1. \(11a+15\) 2. \(4a+2\)

Step 1: Apply the distributive property to the first expression

First, we apply the distributive property to the first expression: \(5(a+2)+6a-7+(8+4)\). We multiply 5 by both terms within the parentheses: $$ 5a+10+6a-7+(8+4) $$

Step 2: Combine like terms for the first expression

Next, combine like terms in the expression we got in step 1: $$ (5a+6a)+(10-7+8+4) $$ This simplifies to: $$ 11a+15 $$

Step 3: Apply the distributive property to the second expression

For the second expression, \((a+3a+2)\), there is no distributive property to apply since there is no term outside of the parentheses that needs to be distributed.

Step 4: Combine like terms for the second expression

Now, combine like terms for the second expression \((a+3a+2)\): $$ (1a+3a)+2 $$ This simplifies to: $$ 4a+2 $$ Both expressions are now simplified using the distributive property and combining like terms: 1. \(11a+15\) 2. \(4a+2\)

Key Concepts

Combining Like Terms

Combining like terms is an essential concept in algebra that helps simplify complex expressions. Like terms are terms within an expression that have the same variable raised to the same power. For example, in the expression \(5a + 6a + 3b\), both \(5a\) and \(6a\) are like terms, as they both have the variable \(a\). However, \(3b\) is not a like term with \(5a\) or \(6a\) because it has a different variable.

When you combine like terms, you're essentially performing addition or subtraction on the coefficients of these terms. Here's how it's done:

• Identify terms in the expression that are like terms. This means finding terms with the same variables and exponents.
• Add or subtract the coefficients of these like terms.
• Rewrite the expression using the new coefficients.

In the exercise provided, combining like terms was demonstrated as follows:
\((5a + 6a)\) combines to become \(11a\), and \((10 - 7 + 8 + 4)\) combines to become \(15\). Similarly, for the expression \((a + 3a + 2)\), like terms \((1a + 3a)\) are combined to form \(4a\). The simplified expressions become easier to work with and interpret.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression as simple as possible. It involves the following steps:

• Use the distributive property if necessary.
• Combine like terms.

Simplifying makes it easier to understand and solve equations.

The given exercise required the simplification of two expressions. For the first, \(5(a + 2) + 6a - 7 + (8 + 4)\), the expression was simplified in multiple stages. First, applying the distributive property resulted in \(5a + 10 + 6a - 7 + (8 + 4)\). The next step involved combining like terms, resulting in a simpler expression of \(11a + 15\). This process reduces the expression to its most basic form, making it clearer and easier to handle.

Similarly, for \((a + 3a + 2)\), no distributive property was needed, but combining like terms \((1a + 3a)\) led to \(4a + 2\), simplifying the problem for ease of use. This approach is vital in algebra and aids in solving equations efficiently.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (such as \(+\), \(-\), \(\times\), and \(\div\)). They represent quantities and relationships that do not contain equality or inequality signs; that's what makes them different from equations and inequalities.

In algebra, expressions like \(5(a+2)+6a-7+(8+4)\) and \((a+3a+2)\) must be interpreted and manipulated according to specific mathematical rules. These involve simplifying by applying the distributive property and combining like terms, which are foundational skills in algebra.

• Variables: Letters representing numbers. In our exercise, \(a\) is the variable.
• Coefficients: Numbers in front of the variables that indicate how many times the variable is multiplied. In \(5a\), 5 is the coefficient.
• Constants: Numbers on their own without any variable attached, such as 8 or 4 in the expression.

These elements come together within expressions to describe things mathematically. Understanding how to manipulate these components by simplifying can make solving real-world problems much more manageable. Expressive calculations allow you to reframe complex problems into simpler, more manageable terms.