Problem 6 - Algebra for College Students

Question

Solve each equation for \(y\). $$\frac{1}{n}=\frac{a}{y}-\frac{w}{a}$$

Step-by-Step Solution

Answer

y = \frac{a^2n}{a + wn}

Step 1: Isolate the term with the variable y

Rewrite the equation to isolate the term with y: \( \frac{1}{n} = \frac{a}{y} - \frac{w}{a} \) Add \( \frac{w}{a} \) to both sides: \( \frac{1}{n} + \frac{w}{a} = \frac{a}{y} \)

Step 2: Find a common denominator

To combine the fractions on the left side, find a common denominator. The least common multiple of n and a is na. Thus, \(\frac{1}{n} + \frac{w}{a} = \frac{a}{y} \) becomes: \(\frac{a}{an} + \frac{wn}{an} = \frac{a}{y} \)

Step 3: Combine the fractions

Combine the fractions on the left side: \( \frac{a + wn}{an} = \frac{a}{y} \)

Step 4: Cross multiply to solve for y

Cross multiply to get rid of the fractions: \( y(a + wn) = a \times an \) Simplify it: \( y(a + wn) = a^2n \)

Step 5: Solve for y

Finally, divide both sides by \( a + wn \): \( y = \frac{a^2n}{a + wn} \)

Key Concepts

Isolating the Variable

When solving equations, you often need to isolate the variable to one side. This helps in focusing on the term you want to solve for. In our example, we have the equation \(\frac{1}{n} = \frac{a}{y} - \frac{w}{a}\).
To isolate the term with \(y\), add \(\frac{w}{a}\) to both sides:
\(\frac{1}{n} + \frac{w}{a} = \frac{a}{y}\).
Once the term with the variable is isolated, it is easier to manipulate the equation to solve for \(y\).
Isolating the variable is a crucial first step.

Common Denominator

To add or subtract fractions, they need a common denominator. This means finding a shared multiple of the denominators involved. In the equation \(\frac{1}{n} + \frac{w}{a} = \frac{a}{y}\), n and a are the denominators.
The least common multiple of n and a is na. Thus, we rewrite the fractions:
\(\frac{a}{an} + \frac{wn}{an}\).
Having a common denominator allows us to combine these fractions easily.

Cross Multiplication

Cross multiplication is a technique for solving equations involving two fractions. It helps to eliminate the fractions. For the equation \(\frac{a+wn}{an} = \frac{a}{y}\), multiply across the equals sign:
\(y(a + wn) = a \times an\).
Cross multiplication transforms the equation into a simpler form that can be manipulated further.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations. After cross-multiplying, we get \y(a + wn) = a^2n\. To solve for \(y\), divide both sides by \(a + wn\):
\(y = \frac{a^2n}{a + wn}\).
This step-by-step rearrangement of terms is key to solving equations accurately.