Problem 100 - Algebra A Combined Function

Question

Match each expression in the first column with its value in the second column. a. \((1+4) \cdot 6-3\) 15 b. \(1+4 \cdot(6-3)\) 13 c. \(1+4 \cdot 6-3\) 27 d. \((1+4) \cdot(6-3)\) 22

Step-by-Step Solution

Answer

a: 27, b: 13, c: 22, d: 15

Step 1: Simplify Expression a

Expression a: \((1+4) \cdot 6 - 3\)First, solve inside the parentheses: \(1+4 = 5\).Then multiply by 6: \(5 \cdot 6 = 30\).Finally, subtract 3: \(30 - 3 = 27\).

Step 2: Simplify Expression b

Expression b: \(1 + 4 \cdot (6 - 3)\)First, solve inside the parentheses: \(6 - 3 = 3\).Then multiply by 4: \(4 \cdot 3 = 12\).Finally, add 1: \(1 + 12 = 13\).

Step 3: Simplify Expression c

Expression c: \(1 + 4 \cdot 6 - 3\)First, perform the multiplication: \(4 \cdot 6 = 24\).Then add 1: \(1 + 24 = 25\).Finally, subtract 3: \(25 - 3 = 22\).

Step 4: Simplify Expression d

Expression d: \((1+4) \cdot(6-3)\)First, solve inside the first parentheses: \(1+4 = 5\).Then solve inside the second parentheses: \(6-3 = 3\).Finally, multiply the results: \(5 \cdot 3 = 15\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental aspect of algebra where we reduce the complexity of expressions using mathematical operations. It involves combining like terms and systematically applying the order of operations to streamline calculations. For instance, take the expression

1. \( (1+4) \cdot 6 - 3 \)

First, add inside the parentheses:

• \(1 + 4 = 5\)

You simplify expressions step by step, always respecting the order of operations. Here, you next multiply by 6:

• \(5 \cdot 6 = 30\)

Finally, you subtract 3 to arrive at:

• \(30 - 3 = 27\)

As you can see, simplifying expressions involves breaking down each step carefully to ensure you arrive at the correct result. This method is crucial in making calculations more manageable and less error-prone.

Arithmetic Operations

Arithmetic operations are the building blocks of math that include addition, subtraction, multiplication, and division. Let's break down how these operations interact in expressions to provide an accurate outcome.
Consider the expression

1. \(1 + 4 \cdot 6 - 3\)

To simplify this, start with multiplication:

• \(4 \cdot 6 = 24\)

Next, add 1:

• \(1 + 24 = 25\)

Finally, subtract 3:

• \(25 - 3 = 22\)

The principles of arithmetic operations dictate that multiplication and division must be handled before addition and subtraction in mixed expressions.

This sequence helps maintain consistent results across different calculations. Understanding how to correctly sequence these operations ensures that expressions are solved accurately, leading to reliable outcomes.

Parentheses in Math

Parentheses in math serve a crucial purpose: they dictate the order in which operations are performed, overruling the standard order of operations. This is often referred to as grouping and can dramatically change the outcome of expressions.
For example, consider

1. \((1+4) \cdot (6-3)\)

First, resolve what is within the parentheses:

• \(1+4 = 5\)
• \(6-3 = 3\)

Then, multiply the simplified results:

• \(5 \cdot 3 = 15\)

Parentheses help you manipulate expressions to focus on specific operations first, ensuring calculations reflect your intended procedure.

When you correctly apply parentheses, you achieve consistent results and accurately represent mathematical ideas. Misplacing or neglecting parentheses often leads to incorrect answers and misunderstandings of the problem at hand. It's vital to respect these symbols to preserve the integrity of solving expressions.