Problem 57 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 9 \cdot 7-4 \cdot 5-3 \cdot 2+4 \cdot 7 $$

Step-by-Step Solution

Answer

The simplified expression is 65.

Step 1: Multiply the Numbers

First, calculate the products in the expression: - Multiply 9 by 7: \(9 \cdot 7 = 63\)- Multiply 4 by 5: \(4 \cdot 5 = 20\)- Multiply 3 by 2: \(3 \cdot 2 = 6\)- Multiply 4 by 7: \(4 \cdot 7 = 28\)

Step 2: Substitute Products into the Expression

Replace the products in the original expression with the results from Step 1: \[63 - 20 - 6 + 28\]

Step 3: Perform Addition and Subtraction from Left to Right

Simplify the expression by performing the addition and subtraction in order:1. Subtract 20 from 63: \(63 - 20 = 43\)2. Subtract 6 from 43: \(43 - 6 = 37\)3. Add 28 to 37: \(37 + 28 = 65\)

Step 4: Final Simplified Expression

The expression simplifies to 65.

Key Concepts

Understanding the Order of Operations

When simplifying numerical expressions, knowing the order of operations is crucial. This order dictates the sequence in which operations should be carried out to ensure accurate results. The standard sequence is remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In this specific exercise, our primary concern is multiplication and then addition or subtraction. It is important to first perform all multiplication operations before moving on to addition and subtraction. This ensures that each calculation builds correctly on the prior calculations, as per the established order of operations.

Mastering Multiplication in Expressions

Multiplication is the process of combining equal groups. In algebra and expressions, multiplication is often done once, so capturing the correct values is key. In numerical simplification exercises:

• Identify multiplication operations.
• Calculate each product.

For example, from the original expression, you multiply 9 by 7 to get 63, 4 by 5 to get 20, and so on. Ensuring precision here is pivotal because mistakes can cascade, affecting all subsequent parts of the problem. By double-checking each multiplication step, we're setting ourselves up for a smoother path towards the final answer.

Proceeding with Addition and Subtraction

After handling multiplication, the next step involves addition and subtraction. These operations are grouped together because they share similar properties and are performed at the same level of priority, moving from left to right. In our example:

• Begin by subtracting the first pair: 20 from 63.
• Continue by subtracting the next number: 6 from 43.
• Finally, add the last number: 28 to 37.

Performing these operations in sequence ensures that the numerical expression simplifies correctly. Each calculation should be verified as you proceed to minimize the risk of errors and maintain the logical flow of arithmetic operations.

Working with Algebraic Expressions

Algebraic expressions, though not identical to our exercise, share some fundamental principles with numerical expressions. They consist of variables, constants, and operation signs to model real-world situations or expressions. Key insights include:

• Like terms should be combined.
• Operations follow the same order: dealing firstly with multiplication.
• Simplification involves carefully applying addition and subtraction to combined constants.

A strong grasp of numerical expression simplification lays the groundwork for tackling more complex algebraic expressions. As you become comfortable with numbers, the transition to variables becomes less daunting. Remember that each step in solving algebraic problems often mirrors those of simpler arithmetic ones; it’s just about keeping track of which unknowns or constants apply throughout.