Question

If \(y=1 / x+x\) and \(x=4\) then \(y\) is: (A) \(17 / 4\) (B) \(4 / 5\) (C) \(5 / 4\) (D) \(17 / 8\) (E) None of the above

Short answer

Option (A) \( \frac{17}{4} \).

Step-by-step solution

- Understand the Problem

We are given the function \( y = \frac{1}{x} + x \) and asked to find the value of y when \( x = 4 \).

- Substitute the Given Value

Substitute \( x = 4 \) into the equation \( y = \frac{1}{x} + x \). This results in \( y = \frac{1}{4} + 4 \).

- Perform the Calculations

Calculate each term: \( \frac{1}{4} \) is 0.25, and adding 4 gives 4.25. Therefore, \( y = 0.25 + 4 = 4.25 \).

- Write the Answer in Fraction Form

Convert 4.25 to a fraction: \( 4.25 = \frac{17}{4} \).

- Match with the Options

Compare \( \frac{17}{4} \) with the given options. The correct answer is (A) \( \frac{17}{4} \).

Key concepts

algebra problem-solving

Algebra problem-solving often requires you to manipulate equations and substitute values. When facing a problem like the one given in the exercise, the first step is to clearly understand what is being asked. For instance, we were given the equation \( y = \frac{1}{x} + x \) and needed to find the value of \( y \) when \( x = 4 \). Start by identifying the relationship between variables. Have a strong grasp of functions and how to substitute variables correctly to find solutions. Breaking the problem down into smaller, manageable steps is key. This helps in simplifying complex problems and improves accuracy.
In algebra, especially in standardized tests like the GMAT, ensure your basics are solid and practice various types of problems to become proficient.

substitution method

The substitution method is a fundamental concept in solving algebraic equations. Here, you replace a variable with a given value to simplify the equation. For this exercise, we substituted \( x = 4 \) into the equation \( y = \frac{1}{x} + x \). Since substitution is often about replacing variables correctly:

• First, write down the equation clearly.
• Identify the variable that needs substitution.
• Replace the variable with the given value.

In this case, substituting \( x \) with 4 gives us \( y = \frac{1}{4} + 4 \). Substitution is effective in progressing towards the solution and is frequently used in algebraic problem-solving.

fraction conversion

Fraction conversion is an essential skill in algebra, especially when dealing with decimal and fractional results. In our exercise, once we calculated the value \( y = 0.25 + 4 \), we needed to convert 4.25 into a fraction. Decimal to fraction conversion involves:

• Determining the place value of the decimal (e.g., 0.25 is in the hundredths place).
• Writing it as a fraction (0.25 = \( \frac{1}{4} \)).

For 4.25, it translates to \( 4 \frac{1}{4} \), which simplifies to an improper fraction \( \frac{17}{4} \). Practicing these conversions can help quickly shift between decimals and fractions, providing more accurate results in tests.

calculation steps

Calculation steps are crucial in solving algebra problems accurately. Each step builds on the previous one. Here’s how the calculation was done for this problem:

• First, understand the equation and the requirement (\( y = \frac{1}{x} + x \), find \( y \) when \( x = 4 \)).
• Next, substitute \( x = 4 \) into the equation: \( y = \frac{1}{4} + 4 \).
• Then, perform the arithmetic calculations: \( \frac{1}{4} \) is 0.25 and add it to 4 which gives 4.25.
• Finally, convert 4.25 into a fraction: 4.25 = \( \frac{17}{4} \).

Following calculation steps methodically helps in ensuring precision and clarity in the final answer. Always double-check each step to avoid errors.