Question

If \(1 / y=1 / x+2 / x\) and \(x=10\), then \(y\) is: A. \(10 / 9\) B. \(10 / 3\) C. \(20 / 9\) D. 8/9 E. None of the above

Short answer

B. \( \frac{10}{3} \)

Step-by-step solution

- Substitute the value of x

Given the equation \( \frac{1}{y} = \frac{1}{x} + \frac{2}{x} \), substitute \( x = 10 \) into the equation.

- Simplify the equation

Replace \( x \) with \( 10 \): \( \frac{1}{y} = \frac{1}{10} + \frac{2}{10} \). Simplify the right-hand side: \( \frac{1}{y} = \frac{1 + 2}{10} = \frac{3}{10} \).

- Solve for y

Rewrite the equation \( \frac{1}{y} = \frac{3}{10} \) to solve for \( y \): \( y = \frac{10}{3} \).

Key concepts

algebra problem solving

Algebra problem solving often involves manipulating equations to find a solution for an unknown variable. In this problem, we are given an equation involving fractions and asked to find the value of one variable when another variable is known. Understanding how to interpret and rearrange the equations is key. By substituting known values and simplifying step-by-step, we can efficiently solve algebraic problems. This systematic approach helps ensure accuracy and clarity in finding the correct solution.

substitution method

The substitution method is a powerful technique in algebra to solve equations where one of the variables is known. It involves replacing the variable with its given value and simplifying the resulting equation. Here’s how it works in this problem:
Given the initial equation \(\frac{1}{y}=\frac{1}{x}+2 \frac{2}{x}\), we substitute \(x = 10\). This substitution is expressed as:

• \(\frac{1}{y}=\frac{1}{10}+\frac{2}{10}\)
• Simplifying the right side, we get \( \frac{1}{y}=\frac{3}{10}\)

By substituting and simplifying, we reduce the complexity step-by-step, making it easier to solve for \(y\).

fraction simplification

Simplifying fractions is essential to solving many algebra problems. Here’s what we did in this exercise:
We started with the equation \( \frac{1}{y}=\frac{1}{10}+\frac{2}{10} \).
To combine these fractions, we made sure to have a common denominator:

• \( \frac{1}{10}+\frac{2}{10}\)
• This combines to \( \frac{1+2}{10} = \frac{3}{10} \).

Once simplified, the equation \( \frac{1}{y} = \frac{3}{10} \) became straightforward to solve by taking the reciprocal:

• \( y = \frac{10}{3} \).

Simplifying fractions and combining them correctly are crucial skills in algebra that help in solving complex equations efficiently.