Problem 52 - Algebra A Combined Function

Question

Simplify each expression. \(4-6(7-3)\)

Step-by-Step Solution

Answer

The simplified expression is 28.

Step 1: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses: \(7 - 3\). Calculate \(7 - 3 = 4\).

Step 2: Multiply

Next, multiply the result from Step 1 by \(-6\) because of the expression \(-6(4)\). Calculate \(-6 \times 4 = -24\).

Step 3: Subtract

Finally, subtract the result from Step 2 from \(4\): \(4 - (-24)\). This becomes \(4 + 24 = 28\).

Key Concepts

Simplification

Simplification of algebraic expressions involves reducing the expression to its simplest form. This means performing all possible arithmetic operations to arrive at a single, more straightforward numerical result. In the context of the exercise, simplification started right away by focusing on an expression wrapped in parentheses.

• Initially, the expression was quite complex: 4-6(7-3).
• By evaluating smaller parts of the expression, it becomes easier to manage and solve.
• Step by step, simplify each part until you cannot simplify further.

Giving priority to operations enclosed in different groupings, including parentheses, is an essential part of simplification in algebra.

Order of Operations

The order of operations is crucial in mathematics to ensure consistent and correct results. The acronym PEMDAS is often used to help remember the order:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

This means that operations inside parentheses must be completed first before any multiplication or subtraction. The exercise demonstrated this by solving the expression inside the parentheses first: (7-3). Only after simplifying this part, multiplication and lastly subtraction were handled. By strictly adhering to this order, you can avoid common mistakes and ensure your solution is correct.

Parentheses in Equations

Parentheses play a critical role in mathematical equations as they dictate the order in which operations should be performed. In our exercise, parentheses signaled the need for initial attention around the subtraction: (7-3).

• The contents of parentheses are tackled before moving on to other arithmetic processes.
• This ensures that calculations proceed in an organized and logical manner.
• Respecting parentheses prevents errors in more complex equations.

Once we determine what's inside the parentheses, we can "release" the confined calculations to integrate them into the expression as a whole. In conclusion, whether dealing with basic arithmetic or advanced equations, parentheses are key to clear and correct mathematics.