Problem 118 - College Algebra

Question

Use the order of operations to simplify each expression. \(\frac{6(-4)-5(-3)}{9-10}\)

Step-by-Step Solution

Answer

The simplified form of the given expression is 9.

Step 1: Handle Parentheses

First, solve the operations within the parentheses: -4 multiplied by 6 equals to -24. Also, -3 multiplied by -5 equals 15.

Step 2: Simplify Numerator

Next, add the results from step 1: -24 plus 15 equals -9.

Step 3: Simplify Denominator

Now, subtract 10 from 9: 9 minus 10 produces -1.

Step 4: Division

Lastly, divide the results of step 2 (-9) by the result of step 3 (-1). This yields 9.

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions is a fundamental aspect of algebra which involves reducing an expression to its most basic form. To achieve this, one must apply the order of operations accurately. In our example, we first address the calculations inside the parentheses such as computing \(6(-4)\) and \(5(-3)\). This is in line with PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) which dictates that operations enclosed in parentheses are given priority.

After the parentheses, we deal with any exponents, which our example lacks. Then we proceed with multiplication and division from left to right, and finally, we tackle addition and subtraction, also from left to right. Simplifying correctly leads to clearer and more concise numerical and algebraic expressions, making it easier to understand and solve mathematical problems.

Understanding how to manipulate and simplify expressions aids not only in solving textbook problems but also in developing a more intuitive grasp of algebraic concepts for future lessons.

Arithmetic Operations

Arithmetic operations form the backbone of many mathematical problems, encompassing addition, subtraction, multiplication, and division. In the given exercise, we encounter a mixture of these operations. We begin with multiplication inside parentheses, followed by addition of those products. Once we have simplified the numerator of our fraction, we move to the subtraction in the denominator.

Something to remember is that when performing arithmetic operations, especially with negative numbers, it's crucial to keep track of the signs, as they can dramatically alter the outcome. For instance, multiplying or dividing two negative numbers results in a positive number—knowledge that was key to simplifying the numerator in our example. Having a solid understanding of how to execute these operations allows for the successful simplification of complex expressions.

College Algebra

College algebra can sometimes seem intimidating, yet it is essentially a study of the properties and applications of operations and relations. This subject builds on high school algebra and integrates more complex problems, strengthening analytical and problem-solving skills. The exercise we've discussed represents a common type of problem in college algebra that requires simplifying expressions.

In essence, college algebra involves exploring a wide range of mathematical concepts, including functions, equations, inequalities, and real-world applications of algebra. By mastering order of operations and simplification techniques, you lay a robust foundation for tackling more advanced concepts. This kind of early exposure ensures that you are well-prepared for the diverse mathematical challenges you will face throughout your academic and professional life.