Problem 37 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ \frac{5 x}{7}+3 x-7, \text { if } x=14 $$

Step-by-Step Solution

Answer

The value of the expression is 45.

Step 1: Substitute the Variable

Substitute the given value of \( x = 14 \) into the expression. So the expression becomes \( \frac{5(14)}{7} + 3(14) - 7 \).

Step 2: Evaluate the Fraction

Calculate the fraction part of the expression: \( \frac{5(14)}{7} \). Since \( 5 \times 14 = 70 \), the fraction becomes \( \frac{70}{7} = 10 \).

Step 3: Calculate Each Term Separately

Evaluate the terms in the expression separately: \( \frac{70}{7} = 10 \), \( 3 \times 14 = 42 \), and \( -7 \).

Step 4: Combine All Terms

Add and subtract the calculated terms: \( 10 + 42 - 7 \).

Step 5: Final Evaluation

First, add \( 10 + 42 = 52 \). Then subtract \( 7 \) from \( 52 \), which results in \( 45 \).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that simplifies expressions by replacing variables with their given numerical values. This method makes complex problems easier to solve:
- Begin by identifying the variable in the algebraic expression.
- Replace or "substitute" the variable with its respective value provided in the problem.

In our case, the exercise asks us to find the value of the expression \( \frac{5x}{7} + 3x - 7 \) when \( x = 14 \):
- Substitute 14 in place of \( x \), turning the expression into \( \frac{5(14)}{7} + 3(14) - 7 \).
- This allows you to work with numerical operations instead of abstract symbols, making the overall solving process more manageable.

Understanding substitution equips you with the ability to convert variable expressions into numerical ones, thus paving the way for further manipulation and simplification of algebraic problems.

Fractions

Fractions represent parts of a whole and are central to various mathematical operations. In our exercise, the fraction \( \frac{5x}{7} \) involves both multiplication and division:
- First, compute the numerator by multiplying 5 and 14, which gives 70.
- Then, divide 70 by the denominator 7. This yields a quotient of 10, simplifying the expression to a whole number.

When working with fractions, it's critical to understand the steps:
- Multiply the numbers in the numerator for expressions that involve variables.
- Perform the division by the number in the denominator.

These steps transition what may seem like a complex fraction into a simple, direct calculation.

Arithmetic Operations

Arithmetic operations, including addition, subtraction, multiplication, and division, form the pillar of evaluating algebraic expressions. Let's break down these operations using our example:
- After substituting and simplifying the fraction, compute each term separately:
- Fraction part: \( \frac{70}{7} = 10 \).
- Multiplication: \( 3 \times 14 = 42 \).
- The constant term \( -7 \).

Next, all these separate results are combined using addition and subtraction:
- First, add the values:
- Combine 10 and 42 to get 52.
- Then, perform the subtraction: subtract 7 from 52 to arrive at the final answer of 45.

Understanding how these basic operations interact within an expression helps solve parts sequentially, avoiding mistakes and achieving accurate results. They are key in making complex expressions straightforward by breaking them down into smaller, manageable steps.