Solving (-9 + -5) X (5t - 56) When T = 9

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Solving (-9 + -5) x (5t - 56) when t = 9

Alright, guys, let's dive into this math problem together! We're tasked with solving the expression (-9 + -5) x (5t - 56) when t = 9. It looks a bit intimidating at first, but don't worry; we'll break it down step by step to make it super easy to understand. Our goal is to simplify the expression by first dealing with the parentheses and then substituting the value of t to get our final answer. So, grab your calculators (or just your brains!), and let's get started!

Step-by-Step Solution

1. Simplify the First Parenthesis

First, let's tackle the first part of the expression: (-9 + -5). This is just adding two negative numbers. Remember, when you add negative numbers, you're essentially moving further to the left on the number line. So, -9 + -5 is the same as -9 - 5. Calculating this gives us:

-9 + -5 = -14

So, we've simplified the first parenthesis, and now our expression looks like this:

-14 x (5t - 56)

2. Substitute the Value of t

Next up, we need to substitute the value of t into the second parenthesis. We're given that t = 9. So, we replace t with 9 in the expression (5t - 56):

(5 * 9 - 56)

3. Simplify the Second Parenthesis

Now, let's simplify the second parenthesis. First, we perform the multiplication:

5 * 9 = 45

So, our expression inside the parenthesis becomes:

(45 - 56)

Now, subtract 56 from 45:

45 - 56 = -11

So, the second parenthesis simplifies to -11. Now our entire expression looks like this:

-14 x -11

4. Final Calculation

Finally, we need to multiply -14 by -11. Remember, when you multiply two negative numbers, you get a positive number. So, we're really calculating 14 x 11:

14 x 11 = 154

Therefore, the final answer to the expression (-9 + -5) x (5t - 56) when t = 9 is 154.

Detailed Explanation of Each Step

Understanding Integer Arithmetic

Let's break down the basics of integer arithmetic to make sure everyone's on the same page. When we deal with positive and negative numbers, it's crucial to remember a few key rules:

  • Adding two positive numbers: This is straightforward. For example, 5 + 3 = 8. You're simply moving to the right on the number line.
  • Adding two negative numbers: As we saw earlier, adding two negative numbers means moving further to the left on the number line. For example, -5 + -3 = -8.
  • Adding a positive and a negative number: This is like moving right and then left (or vice versa) on the number line. For example, 5 + -3 = 2 (move 5 to the right, then 3 to the left) and -5 + 3 = -2 (move 5 to the left, then 3 to the right).
  • Subtracting a positive number: This is the same as adding a negative number. For example, 5 - 3 = 5 + -3 = 2.
  • Subtracting a negative number: This is the same as adding a positive number. For example, 5 - -3 = 5 + 3 = 8. Subtracting a negative is like taking away a debt, which increases your total.
  • Multiplying or dividing two positive numbers: The result is positive. For example, 5 * 3 = 15 and 15 / 3 = 5.
  • Multiplying or dividing two negative numbers: The result is positive. For example, -5 * -3 = 15 and -15 / -3 = 5.
  • Multiplying or dividing a positive and a negative number: The result is negative. For example, 5 * -3 = -15 and -15 / 3 = -5.

These rules are fundamental to solving any expression involving integers, so make sure you're comfortable with them.

The Order of Operations (PEMDAS/BODMAS)

In mathematics, we follow a specific order of operations to ensure that we arrive at the correct answer. This order is often remembered by the acronyms PEMDAS or BODMAS:

  • Parentheses / Brackets: First, simplify anything inside parentheses or brackets.
  • Exponents / Orders: Next, evaluate any exponents or orders (powers and square roots).
  • Multiplication and Division: Then, perform multiplication and division from left to right.
  • Addition and Subtraction: Finally, perform addition and subtraction from left to right.

Following this order ensures that we all solve the same expression in the same way, leading to a consistent and correct result. In our problem, we adhered to this order by first simplifying the expressions within the parentheses before performing any multiplication.

Common Mistakes to Avoid

When solving expressions like this, it's easy to make small mistakes that can lead to the wrong answer. Here are some common pitfalls to watch out for:

  • Incorrectly adding negative numbers: Double-check your signs when adding negative numbers. Remember, -5 + -3 is -8, not 2.
  • Forgetting the order of operations: Always follow PEMDAS/BODMAS. If you multiply before simplifying the parentheses, you'll get the wrong answer.
  • Sign errors in multiplication: Remember that a negative times a negative is a positive, and a negative times a positive is a negative.
  • Misinterpreting subtraction: Remember that subtracting a negative is the same as adding a positive. 5 - -3 is 8, not 2.
  • Rushing through the steps: Take your time and double-check each step to avoid careless errors. Math requires precision!

By being mindful of these common mistakes, you can increase your accuracy and confidence in solving mathematical expressions.

Alternative Approaches

While we solved this problem step-by-step, there are sometimes alternative approaches you could take. However, in this case, the step-by-step method is the most straightforward and least prone to error. For example, you could distribute the -14 across the terms in the second parenthesis before substituting t = 9, but that would likely make the problem more complicated than it needs to be.

In general, it's best to simplify within parentheses first and then substitute variable values. This minimizes the risk of making mistakes and keeps the calculations manageable.

Real-World Applications

You might be wondering,