Problem 32 - Elementary and Intermediate Algebra

Question

To prepare for Section 9.4, review order of operations \((\text {Section } 1.8)\). $$ \begin{aligned} 6(2 \cdot 7-3(-4))-4(3(-8)-10) \\ +5(4 \cdot 3-(-2) 7) \end{aligned} $$

Step-by-Step Solution

Answer

422

Step 1: Simplify Inside the Parentheses

Start by simplifying the expressions inside the parentheses. Calculate the values for each operation inside every set of parentheses: 1. Inside the first parentheses: \(2 \times 7 - 3(-4)\) 2. Inside the second parentheses: \(3(-8) - 10\) 3. Inside the third parentheses: \(4 \times 3 - (-2) \times 7\)

Step 2: Perform the Calculations

Compute the individual operations step-by-step: 1. \(2 \times 7 = 14\) and \(-3 \times -4 = 12\), so the expression inside the first parentheses becomes \(14 + 12 = 26\) 2. \(3 \times -8 = -24\) and \(-24 - 10 = -34\), so the expression inside the second parentheses remains \(-34\) 3. \(4 \times 3 = 12\) and \(-2 \times 7 = -14\), so the expression inside the third parentheses becomes \(12 + 14 = 26\)

Step 3: Substitute Back and Simplify

Substitute the simplified expressions back into the main equation: \(6(26) - 4(-34) + 5(26)\)

Step 4: Multiply and Simplify Further

Multiply out the terms and combine them: 1. \(6 \times 26 = 156\) 2. \(-4 \times -34 = 136\) 3. \(5 \times 26 = 130\)

Step 5: Final Calculation

Combine the results of the multiplications: \(156 + 136 + 130 = 422\)

Key Concepts

simplify inside parentheses

Before diving into the main calculations, you must simplify the expressions inside the parentheses first. This includes handling multiplications and subtractions present within them. For example, consider the term inside the first set of parentheses: \(2 \times 7 - 3(-4)\). Here, you need to:

• Multiply \(2\) and \(7\)
• Multiply \(3\) by \(-4\)

Then, combine the results of these operations to simplify the expression inside the parenthesis. Apply this method to other parentheses as well.

multiplication

Once you've simplified the contents of the parentheses, it’s time to take care of the multiplications. For example: After simplifying, your equations inside the parentheses might look like: \(14 + 12\), \(-24 - 10\), and \(12 + 14\). Now compute each:

• First parentheses becomes \(26\)
• Second parentheses becomes \(-34\)
• Third parentheses becomes \(26\)

This step is crucial to move on to the next part of the solution.

substitution

Once the expressions inside the parentheses are simplified, substitute these values back into the main equation. For instance, if you had \(6(2 \times 7 - 3(-4)) - 4(3(-8) - 10) + 5(4 \times 3 - (-2) \times 7)\), after simplification, it becomes \(6(26) - 4(-34) + 5(26)\). Now you can clearly see the simplified numbers placed back in the original spots.

addition

With all the simplified values substituted back, the next step involves handling any remaining addition or subtraction. For example, you may initially get terms like \(6(26)\), \(-4(-34)\), and \(5(26)\). The goal is to multiply these values first and collect the results:

• \(6 \times 26 = 156\)
• $$-4 \times -34 = 136$$
• \(5 \times 26 = 130\)

So you get \(156\), \(136\), and \(130\) to work with.

final calculation

The final calculation combines all the results of previous operations into one tidy sum. After obtaining \(156\), \(136\), and \(130\), add them together to get the final answer:

• \(156 + 136 + 130 \rightarrow 156 + 136 = 292\)
• \(292 + 130 = 422\)

Thus, the final answer to the original expression \(6(2 \times 7 - 3(-4)) - 4(3(-8) - 10) + 5(4 \times 3 - (-2) \times 7)\) is \(422\).