Problem 29 - Elementary Algebra

Question

Simplify. $$ 6-3(a-2 b)+7(5 a-3 b) $$

Step-by-Step Solution

Answer

The simplified expression is \(32a - 15b + 6\).

Step 1: Distribute -3 into the First Parenthesis

Start by distributing the \(-3\) into the terms inside the parenthesis \(a - 2b\). This operation becomes: \[6 - 3a + 6b\]

Step 2: Distribute 7 into the Second Parenthesis

Next, distribute the 7 into the terms inside the second parenthesis \(5a - 3b\). This operation results in:\[35a - 21b\]

Step 3: Combine Like Terms

Add together the terms from both steps along with the initial constant term:\[6 - 3a + 6b + 35a - 21b\]Now, group and combine the like terms. Combine \(35a - 3a\) and \(6b - 21b\):\[6 + 32a - 15b\]

Step 4: Simplified Expression

The simplified expression of the entire equation is:\[32a - 15b + 6\].

Key Concepts

Distribution Property

The distribution property, also known as the distributive law, is a fundamental concept in algebra helping to simplify expressions and perform multiplication effectively. It involves distributing a multiplication operation over addition or subtraction inside parentheses. For example, if you have a term like \( c(a + b) \), you can distribute \( c \) to both \( a \) and \( b \), resulting in \( ca + cb \). This is useful for breaking down complex expressions into simpler components.
In our exercise, we applied the distribution property to two different sets of terms:

• First, we distributed \(-3\) across \( (a - 2b) \). This changed our expression to \(-3a + 6b\).
• Second, we distributed \(7\) over \( (5a - 3b) \), giving us \(35a - 21b\).

Using the distribution property makes the expression easier to work with further.

Combining Like Terms

Combining like terms is an essential skill in simplifying algebraic expressions. It involves grouping terms that have the same variables raised to the same power and then adding or subtracting the coefficients of these terms. Only terms with exactly the same variable part can be combined.
Here’s how it works:

• Consider our expression after distribution: \(-3a + 6b + 35a - 21b\).
• Among these, \(-3a\) and \(35a\) are like terms because they contain the same variable \(a\). Similarly, \(6b\) and \(-21b\) are like terms because they share the variable \(b\).
• We combined \(-3a + 35a\) to get \(32a\), and \(6b - 21b\) to get \(-15b\).

Combining like terms reduces the complexity of the expression, preparing it for further steps, like solving equations.

Simplifying Expressions

Simplifying expressions involves several steps to reduce an algebraic expression to its simplest form. This process often includes distributing values into parentheses, combining like terms, and arranging terms in a standard mathematical order.
The key steps in simplifying an expression include:

• Applying distribution to eliminate parentheses and multiply terms.
• Combining like terms to condense similar items into a single term.
• Ordering terms in a conventional way, typically starting with high powers of variables and moving to constants.

In our example expression \(32a - 15b + 6\), the process concluded with identifying each type of term, ensuring no further simplification was possible.
Simplifying expressions makes solving equations and substituting values in for variables more straightforward.