Problem 73 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ \frac{3 \cdot 8-4 \cdot 3}{5 \cdot 7-34}+19 $$

Step-by-Step Solution

Answer

31

Step 1: Simplify the Numerator

First, simplify the expression in the numerator: \( 3 \cdot 8 - 4 \cdot 3 \). This involves two multiplications and a subtraction.Calculate \( 3 \cdot 8 = 24 \) and \( 4 \cdot 3 = 12 \).Then, subtract: \( 24 - 12 = 12 \).

Step 2: Simplify the Denominator

Next, simplify the expression in the denominator: \( 5 \cdot 7 - 34 \).Calculate \( 5 \cdot 7 = 35 \).Then, subtract: \( 35 - 34 = 1 \).

Step 3: Divide the Fraction

Now take the simplified numerator and denominator to divide: \( \frac{12}{1} = 12 \).

Step 4: Add to 19

Finally, add the result from the division to 19: \( 12 + 19 = 31 \).

Key Concepts

Simplifying Expressions

Simplifying expressions means breaking down mathematical expressions into their simplest form, making calculations easier to perform. This kind of simplification is crucial when dealing with complex equations.

• Always start with operations inside parentheses or brackets if there are any.
• Proceed by evaluating any exponents.
• Simplify by conducting multiplication and division from left to right.
• Finally, perform addition and subtraction, also from left to right.

Be cautious about the order of operations; it's like following a "recipe" for math. This methodology helps in minimizing errors when simplifying larger expressions.

Arithmetic Operations

Arithmetic operations form the basis of all mathematical calculations. These include addition, subtraction, multiplication, and division. Each operation serves as a building block for solving expressions.

• Addition: Combining two or more numbers to get a total sum.
• Subtraction: Finding the difference between numbers by taking one away from another.
• Multiplication: Increasing the value of a number by a specified number of times, often represented as repeated addition.
• Division: Splitting a number into equal parts or groups.

Understanding these operations is crucial for handling more complex equations and for simplifying expressions effectively.

Fraction Operations

Handling fractions involves understanding how to operate with the numerator and the denominator effectively. To simplify expressions involving fractions, one must be adept with fraction operations.

• Addition and Subtraction: Ensure fractions have the same denominator before combining them.
• Multiplication: Multiply the numerators together and the denominators together.
• Division: Multiply by the reciprocal of the fraction you are dividing by.

In our exercise, the fraction is simplified by ensuring the numerator and denominator perform necessary arithmetic operations before division, enabling simpler calculations and easier final results.