Problem 2 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ 9(4-2)+6(8+2)-3(1+4) $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: 9(4-2)+6(8+2)-3(1+4) Answer: 63

Step 1: Simplify inside parentheses

First, we will simplify the expression inside each parenthesis. $$ 9(4-2)+6(8+2)-3(1+4) $$ $$ 9(2)+6(10)-3(5) $$

Step 2: Perform multiplication

Next, we will perform the multiplication operations. $$ 9\times2+6\times10-3\times5 $$ $$ 18+60-15 $$

Step 3: Perform addition and subtraction

Finally, we will perform addition and subtraction operations from left to right. $$ 18+60-15 $$ $$ 78-15 $$ $$ 63 $$ The simplified expression is 63.

Key Concepts

Understanding Parentheses in Order of Operations

When dealing with mathematical expressions, parentheses play a crucial role in guiding us on which part of the expression to deal with first. To simplify an expression accurately, always start with operations inside parentheses. They tell us that any expressions contained within them should be evaluated as a priority.
For example, consider the expression:

• 9(4-2)+6(8+2)-3(1+4)

In this expression, you need to focus on simplifying

• inside (4-2)
• inside (8+2)
• and inside (1+4)

By calculating these, you get a simplified form:

• 9(2)+6(10)-3(5)

Now, you can carry on with the subsequent steps, but only after this part is complete. Parentheses help in chunking the problem into more manageable pieces, maintaining the accuracy of the expression as we move ahead.

The Role of Multiplication in Simplifying Expressions

Once you have resolved the parentheses in your expression, the next operation to look at is multiplication. This step follows because multiplication takes precedence before addition and subtraction in standard mathematical rules. This ensures that your calculation follows the correct order. Continuing with the given problem:

• 9(2) becomes 9 times 2
• 6(10) becomes 6 times 10
• 3(5) becomes 3 times 5

Calculating these gives us:

• 18 + 60 - 15

Multiplication acts as a foundation to simplify the expression further. When done correctly and in order, it makes the progression to the next steps smoother and more direct.

Addition and Subtraction: Getting to the Final Result

The final part of simplifying your expression involves performing addition and subtraction. Once you have carried out multiplication, only then should you move on to these steps. Remember to work from left to right, which is the standard approach. From our earlier calculations:

• We had the expression: 18 + 60 - 15

Follow these steps:

• Start by adding 18 and 60 to get 78
• Then, subtract 15 from 78, which results in 63

By taking these small steps and ensuring you follow the order of operations, the complex initial expression is simplified to a simple final answer of 63. Keeping addition and subtraction at the end maintains the structural integrity of your expression until every other operation is dealt with first.