Problem 70 - Algebra for College Students

Question

Simplify each numerical expression. $$7(8-9)+(-6)(4)$$

Step-by-Step Solution

Answer

The simplified expression is -31.

Step 1: Simplify inside the parentheses

Start by simplifying the expression inside the parentheses. We have \((8 - 9)\), which simplifies to \(-1\). So, the expression becomes \(7(-1) + (-6)(4)\).

Step 2: Multiply the terms

Next, multiply the terms outside the parentheses. For \(7(-1)\), the multiplication yields \(-7\). For \((-6)(4)\), it results in \(-24\). Now the expression is \(-7 + (-24)\).

Step 3: Add the terms

Finally, add the two terms together. Adding \(-7\) and \(-24\) gives us \(-31\). So the simplified expression is \(-31\).

Key Concepts

Order of Operations

When you're simplifying a numerical expression, following the order of operations is essential. It's like a roadmap, guiding you through which calculations to perform first. The order of operations can be remembered by the acronym PEMDAS.

• P: Parentheses first
• E: Exponents (or powers, roots)
• M/D: Multiplication and Division (left to right)
• A/S: Addition and Subtraction (left to right)

Why is this order important? Well, if everyone didn't follow the same set of rules, you'd end up with different solutions. That could be very confusing! Stick to the order and you'll always get the correct result. Starting with parentheses is crucial, as it sets the stage for all subsequent calculations.

Parentheses

Parentheses are a huge priority in math expressions because they indicate which operations to perform first. In our example, we see \((8 - 9)\).
Working within the parentheses first, we simplify this to \(-1\).

Here's why parentheses matter:

• They can change the value of an expression dramatically.
• They help group parts of expressions, providing clarity.

Remember, always solve what’s inside the parentheses before doing anything else. It's the start of breaking down a complex problem into an easier one.

Multiplication

After simplifying the content inside parentheses, we move on to multiplication. It's a straightforward but very important part of the process.
In our expression, after addressing the parentheses, we're left with \(7(-1)\) and \((-6)(4)\).
Multiplication involves calculating the product of numbers:

• Step 1: Multiply 7 by \(-1\) to get \(-7\).
• Step 2: Multiply \(-6\) by 4 to get \(-24\).

When multiplying, pay attention to the signs:

• When you multiply a positive and a negative number, you always get a negative product.
• When you multiply two negatives, the product is positive. However, our example doesn't have two negatives multiplied together, so all our products are negative.

Negative Numbers

Negative numbers might seem daunting at first, but they follow a consistent set of rules. Mastering these rules simplifies any expression involving them. Interacting with negative numbers often appears in subtraction, addition, and multiplication.

In our expression, after evaluating the multiplications, we need to add \(-7\) and \(-24\).

Both numbers are negative, so you want to think about moving in the same direction along the number line.

Combine their absolute values (7 and 24), which equals 31.

Keep the negative sign, since both nums are negative.

The sum thus becomes \(-31\).Managing negative numbers involves consistent application of these principles for reliable results.