Problem 76 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate \(\frac{x-y}{3}\) when \(x\) is 2 more than 5 times \(y\) and \(y=4\)

Step-by-Step Solution

Answer

The value of the expression \(\frac{x - y}{3}\) when \( x \) is 2 more than 5 times \(y\) and \(y\) is 4 is \(6\).

Step 1: Identify Given Values

We are given that \( y = 4 \) and the value of \( x \) is 2 more than 5 times \( y \). So, we substitute the value of \( y \) in the given equation.

Step 2: Substitute Value of y in x

The given relationship between \( x \) and \( y \) can be written as \( x = 5y + 2 \). Now, inserting \( y = 4 \) into this equation, it becomes \( x = 5*4 + 2 \) which simplifies to \( x = 22 \).

Step 3: Substitute Values of x and y in Original Expression

The initial expression was \(\frac{x - y}{3}\). Substitute \( x = 22 \) and \( y = 4 \) into this equation. It becomes \(\frac{22 - 4}{3}\) which simplifies to \(6\).

Key Concepts

Expressions

Expressions in algebra are like sentences in language. They contain numbers, variables, and arithmetic operations. An expression can be simple, like a single number or variable, or it can be complex with multiple terms and operations.

In our exercise, the expression is \( \frac{x-y}{3} \). It tells us to perform the operations of subtraction and division on the terms \( x \) and \( y \). Expressions do not have an equality or inequality symbol, meaning they cannot "stand alone" as a full equation.

Understanding expressions is crucial in algebra. You must know how to manipulate them through operations like addition, subtraction, multiplication, and division. This gives you the toolset to solve problems and evaluate different scenarios.

Substitution method

The substitution method is a key algebraic technique used to replace variables with numbers or other expressions. It helps us evaluate expressions or solve equations by simplifying them step by step.

In our task, we started with the condition \( y = 4 \), allowing us to substitute \( y \) with 4. Next, according to the problem, \( x \) is defined as "2 more than 5 times \( y \)." So we replace \( y \) in this formula with 4, leading us to the equation \( x = 5 \times 4 + 2 \). This simplifies to \( x = 22 \).

When substitution allows us to express every variable in terms of numbers or constants, it makes the task of evaluating or solving much easier and straightforward.

Simplification

Simplification in algebra involves reducing expressions to their simplest form. This process helps to make expressions easier to understand and quicker to solve.

After substituting the values of \( x \) and \( y \) into the expression \( \frac{x-y}{3} \), we simplified it further. First, the calculation \( x - y \) translates to \( 22 - 4 = 18 \). Then, dividing 18 by 3 gives the final result of 6.

When simplifying, remember:

• Perform operations according to the order of operations (parentheses, exponents, multiplication/division, addition/subtraction).
• Keep track of negative signs and fractions accurately.
• Always check your work to ensure no steps were skipped or mistakes made in arithmetic.

Simplification makes complex problems manageable and leads to clear, concise answers.