Problem 54 - Algebra A Combined Function

Question

Solve each equation for the indicated variable. $$ \frac{1}{5}+\frac{2}{y}=\frac{1}{x} \text { for } x $$

Step-by-Step Solution

Answer

\(x = \frac{5y}{y + 10}\)

Step 1: Clear Fraction Denominators

To eliminate the fractions, find a common denominator. The least common multiple of the denominators 5, \(y\), and \(x\) is \(5xy\). Multiply every term of the equation by \(5xy\): \(5xy \left( \frac{1}{5} + \frac{2}{y} = \frac{1}{x} \right)\).

Step 2: Distribute and Simplify

Distribute \(5xy\) across each term: \(xy + 10x = 5y\).

Step 3: Isolate the Term with x

Move \(10x\) to the right side of the equation: \(xy = 5y - 10x\).

Step 4: Solve for x

To solve for \(x\), divide every term by \(y\): \(x = \frac{5y}{y} - \frac{10x}{y}\). Simplify to find \(x = \frac{5 - 10x}{y}\).

Step 5: Rearrange to Isolate x

Move terms involving \(x\) on one side to isolate \(x\): \(x + \frac{10x}{y} = 5\).

Step 6: Factor x out of the Equation

Factor \(x\) out of the left side: \(x \left( 1 + \frac{10}{y} \right) = 5\).

Step 7: Solve for x by Division

Divide both sides by \(1 + \frac{10}{y}\): \(x = \frac{5}{1 + \frac{10}{y}}\).

Step 8: Simplify the Expression

Simplify the denominator: \(x = \frac{5y}{y + 10}\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of an unknown variable that makes the equation true. In the equation given, we need to isolate the variable \( x \) to solve for it. Here's a quick guide to get you started:

• Identify the variable you need to solve for—in this case, \( x \).
• Use algebraic operations to isolate the variable on one side of the equation.
• Simplify each step, keeping track of your operations carefully.

Solving equations may involve several steps like combining like terms, using inverse operations such as addition or subtraction, and sometimes even factoring. The key is to keep the equation balanced by doing the same operation to both sides.

Fractions in Equations

Fractions in equations can be intimidating, but they are just another type of number. You can tackle them effectively with a few strategies:

• Understand that fractions represent division.
• To eliminate fractions, consider multiplying by a common denominator.
• Use the distributive property to handle fractional coefficients.

For instance, in our current equation, fractions are cleared by multiplying through by the least common multiple of all denominators involved. This transforms the equation into a simpler form without fractions, making it more straightforward to work with.

Common Denominators

A common denominator is essential when dealing with fractions in equations. It allows you to eliminate fractions and work with whole numbers. Here's why it's important:

• It simplifies comparison and calculation between fractions.
• Using a common denominator helps in clearing fractions by transforming them into integers.
• The least common multiple (LCM) is typically used as the common denominator.

In our exercise, the denominators 5, \( y \), and \( x \) were involved. By using \( 5xy \) as a common denominator, we multiplied all terms by this value, effectively removing the fractions and simplifying the equation. Such an approach is particularly helpful in algebra as it simplifies complex tasks.