Problem 66 - Algebra 2

Question

Solve each equation. Check your solution. $$ 3(5 a+6 b)+8(2 a-b) $$

Step-by-Step Solution

Answer

The simplified expression is \( 31a + 10b \).

Step 1: Expand the First Term

Begin by expanding the first term in the expression: \[ 3(5a + 6b) = 15a + 18b \]

Step 2: Expand the Second Term

Now, expand the second term in the expression: \[ 8(2a - b) = 16a - 8b \]

Step 3: Combine Like Terms

Add the expanded terms together by combining like terms:\[(15a + 18b) + (16a - 8b) = (15a + 16a) + (18b - 8b)\]Simplify this to:\[31a + 10b\]

Step 4: Final Expression

The simplified expression is:\[ 31a + 10b \]

Step 5: Solution Verification

To verify, substitute simple values for \(a\) and \(b\) to ensure both sides of the original and simplified expressions match. Let's use \(a=1\) and \(b=1\):Original expression:\[ 3(5(1) + 6(1)) + 8(2(1) - 1) = 3(11) + 8(1) = 33 + 8 = 41 \]Simplified expression:\[ 31(1) + 10(1) = 31 + 10 = 41 \]Both calculations give the same result.

Key Concepts

Expansion of Expressions

In algebra, the expansion of expressions involves distributing terms inside parentheses to eliminate them. This process helps to break down complex equations into simpler parts. In our example,:

• First, we expanded \( 3(5a + 6b) \) to become \( 15a + 18b \).
• Then, we expanded \( 8(2a - b) \) to turn it into \( 16a - 8b \).

It's important to multiply each term inside the parentheses by the number outside it. This step is crucial in making equations easier to handle. Using distributive laws here, we ensure that every term inside the parenthesis is accounted for, paving the way for further simplification.

Like Terms

In algebraic expressions, like terms are those with the same variable component raised to the same power. Recognizing these is vital for the next steps. In our example, you should focus on:

• Identifying terms like \( 15a \) and \( 16a \) as like terms because they both contain the same variable, \( a \).
• Similarly, \( 18b \) and \( -8b \) are like terms because they both have the variable \( b \).

Like terms can be combined because they represent the same units within the problem. By adding or subtracting their numerical coefficients, you can simplify your expression further. Recognizing and combining like terms is a simple yet powerful tool to streamline equations.

Simplification

Simplification is the process of reducing an expression to its simplest form. This involves combining like terms as seen earlier. For instance, following expansion:

• Combine \( 15a + 16a \) to make \( 31a \).
• Combine \( 18b - 8b \) to create \( 10b \).

Putting these together gives \( 31a + 10b \). Simplification reduces complexity and helps in solving equations promptly. It makes dealing with expressions more manageable and reduces the potential for errors.

Verification of Solutions

Verification is an essential process to ensure your simplification and solution are correct. You want both the original and simplified expressions to give the same result for chosen values. Here's how:

• Select example values, such as \( a = 1 \) and \( b = 1 \).
• Substitute these into both the original expression and the simplified expression.
• Evaluate both to check they yield the same result.

In our case, both the original expression and simplified expression equaled 41, confirming our approach was correct. Verification not only builds confidence in solutions but also minimizes errors before finalizing answers.