Problem 76 - Algebra for College Students

Question

Simplify each numerical expression. $$-65 \div 5-(-13)(-2)+(-36) \div 12$$

Step-by-Step Solution

Answer

-42

Step 1: Simplify the Division

Begin with the division operations first, starting with \(-65 \div 5\).Divide -65 by 5 to get -13.

Step 2: Simplify the Multiplication

Now simplify the multiplication part: \((-13)(-2)\).Multiply -13 by -2 to get 26.

Step 3: Simplify the Second Division

Next, simplify the second division: \(-36 \div 12\).Divide -36 by 12 to get -3.

Step 4: Substitute and Simplify

Substitute the simplified results back into the expression: \(-13 - 26 - 3\).Calculate the result in the order they appear from left to right.\(-13 - 26 = -39\) and then\(-39 - 3 = -42\).

Key Concepts

Numerical Expressions

Numerical expressions are mathematical phrases that contain numbers and operation signs, but no equal sign. These expressions need to be simplified or evaluated in order to interpret the result fully.
In algebra, simplifying a numerical expression typically involves performing arithmetic operations in the correct order. This includes addition, subtraction, multiplication, and division. The order in which these operations are performed is crucial and is governed by the BIDMAS/BODMAS rule: Brackets, Indices/Orders (such as powers and square roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
By following this order, you ensure that every part of the expression is handled properly, leading to accurate results. In our given exercise, the expression is: \(-65 \div 5-(-13)(-2)+(-36) \div 12\). You should identify and handle each operation in accordance with these rules.

Division in Algebra

Division is an essential operation in algebra, often necessary when simplifying expressions. In the exercise, we have two division operations: \(-65 \div 5\) and \(-36 \div 12\).
Division involves finding how many times a number (the divisor) is contained in another number (the dividend). The process is straightforward:

• For \(-65 \div 5\), divide -65 by 5 to get -13.
• For \(-36 \div 12\), divide -36 by 12 to obtain -3.

Both results are key to simplifying the entire problem.
Keep in mind, when dividing negative numbers, the rule of signs applies where a negative divided by a positive yields a negative result. This rule helps maintain the accuracy of your solution.

Multiplication in Algebra

Multiplication in algebra is crucial when simplifying expressions that involve products of numbers or variables. In the exercise, the multiplication operation to perform is \((-13)(-2)\).
Here's how it works:

• Multiplying -13 by -2 involves the rule that the product of two negative numbers is positive. This results in 26.

The rule of signs plays a critical role here, as it dictates whether the product will be negative or positive based on the signs of the multiplicands.
This understanding ensures clear and correct calculations when performing multiplication. It's important to apply these rules carefully to avoid mistakes, especially when multiple negative signs are involved.

Step by Step Solutions

Step by step solutions are a systematic approach to solving algebraic expressions, making the process easier to understand. For the given problem, we sequentially address each operation:
1. **Simplify the division**: Begin with dividing \(-65 \div 5\) getting -13.
2. **Handle the multiplication**: Compute \((-13)(-2)\) yielding 26.
3. **Process the second division**: Calculate \(-36 \div 12\) to obtain -3.
4. **Substitute and simplify**: Place these results back into the expression: \(-13 - 26 - 3\).
5. **Evaluate step-by-step**: First, calculate \(-13 - 26 = -39\), then \(-39 - 3 = -42\).
This approach navigates through complex expressions one operation at a time, inviting clarity and methodical solving. Crucially, it illustrates the importance of sequence and accuracy in mathematics.