Problem 47 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ -4(1-8)+3(10-3) $$

Step-by-Step Solution

Answer

The result is 49.

Step 1: Simplify Inside the Parentheses

First, evaluate the expressions inside each set of parentheses. Start with the first set: \(1 - 8\). This equals \(-7\). Then evaluate the second set: \(10 - 3\). This equals \(7\).

Step 2: Apply Multiplication

Now substitute the results from Step 1 into the expression. It becomes \(-4(-7) + 3(7)\). Perform the multiplication: \(-4\times(-7) = 28\) and \(3\times 7 = 21\).

Step 3: Add the Results

After multiplying, you're left with adding the two results together. So, compute \(28 + 21\). This gives us \(49\).

Step 4: Verify Using Calculator

Use a calculator to verify the expression step by step: 1. Calculate \(1 - 8 = -7\) and \(10 - 3 = 7\).2. Multiply \(-4\times(-7) = 28\) and \(3\times 7 = 21\).3. Finally, add \(28 + 21 = 49\). The calculator should confirm these results.

Key Concepts

Parentheses Evaluation

Understanding how to evaluate expressions within parentheses is an essential skill in algebra. Parentheses are used to alter the normal sequence of operations and represent groupings in mathematical expressions. Basically, any operations inside parentheses must be performed first before any other operations. This is according to the established mathematical order of operations known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
For example, when looking at the expression \(1 - 8\), you calculate inside the parentheses first, resulting in \(-7\). Similarly, for \(10 - 3\), you compute within the parentheses to get \(7\). By resolving each of these groups first, you ensure that your computations are accurate and align with standard mathematical principles.
Remember, always work from the innermost parentheses outward if multiple sets of parentheses are involved.

Multiplication

After solving what's inside the parentheses in any mathematical expression, multiplication typically follows next. This step involves taking the results from the parentheses and performing products with the coefficients or numbers outside. In our expression, \(-4(-7) + 3(7)\), you'll substitute what's inside the parentheses with the answers obtained: \(-7\) and \(7\).
Let's break this down:

• Multiply \(-4\) by \(-7\) which results in \(28\). Here, recall that multiplying two negative numbers gives a positive product.
• Then, \(3\) times \(7\) results in \(21\).

By doing each multiplication segment separately, you can avoid common pitfalls, like incorrect signs or miscalculations that often occur in more complex expressions.
This method reinforces the importance of focus and precision during mathematical operations.

Addition

Once multiplication is completed, the next step in simplifying an expression is often addition or subtraction. These final operations combine the results from your multiplications to reach a final number. In our case, the expression simplifies to adding \(28\) and \(21\). Simply sum these two numbers:

• Add \(28 + 21\), which results in \(49\).

Addition involves combining numbers; so, ensure accuracy by verifying that all previous calculations were correct and properly executed.
This step illustrates how smaller operations connect to form a complete solution, signifying how key each phase is in mathematical problem-solving.

Calculator Verification

Using a calculator to verify your solution is a useful step in confirming accuracy, especially when dealing with more complex expressions. The calculator steps are straightforward and serve as a dependable cross-check for your manual computations. Here is how you can verify our expression step-by-step:

• Start by calculating within the parentheses first: \(1-8 = -7\) and \(10-3 = 7\).
• Next, perform the multiplications: Calculate \(-4(-7) = 28\) and \(3(7) = 21\).
• Finally, add the results: \(28 + 21 = 49\).

If the calculator confirms what you derived manually through the step-by-step process, you can be assured of your solution's correctness. This is why calculators are invaluable in verifying lengthy calculations, preventing small errors from affecting the overall outcome.