Problem 12 - Algebra for College Students

Question

$$\text { Solve each formula for the indicated variable.}$$ $$\frac{2}{x}=\frac{3}{y}-w \text { for } y$$

Step-by-Step Solution

Answer

y = \( \frac{3}{\frac{2}{x} + w} \)

Step 1: Isolate the Fraction Containing y

Start by isolating the fraction \(\frac{3}{y}\) on one side of the equation. Add \(w\) to both sides of the equation:\[\frac{2}{x} + w = \frac{3}{y}\]

Step 2: Invert Both Sides

Invert both sides of the equation to solve for \(y\). Use the property that if \(a = b\), then \( \frac{1}{a} = \frac{1}{b} \):\[\frac{1}{\frac{2}{x} + w} = \frac{y}{3}\]

Step 3: Solve for y

Multiply both sides by \(3\) to isolate \(y\):\[y = \frac{3}{\frac{2}{x} + w}\]

Key Concepts

Isolating Variables

When solving equations, a crucial step is to isolate the variable you are solving for. In this exercise, we need to solve for the variable \(y\). To do this, we begin by rearranging the terms of the equation.To isolate the fraction \(\frac{3}{y}\), we add \(w\) to both sides of the given equation \(\frac{2}{x} = \frac{3}{y} - w\). This results in \(\frac{2}{x} + w = \frac{3}{y}\). By performing this step, we effectively shift any terms not containing \(y\) to the other side, making it easier to isolate the variable.The main idea here is that we want the variable we are solving for to be alone on one side of the equation. This requires adding, subtracting, multiplying, or dividing both sides of the equation by the necessary values to isolate the variable.

Inverting Fractions

Inverting fractions is a powerful tool in solving equations, especially when dealing with variables in the denominator. In our exercise, after isolating \(\frac{3}{y}\), we reach the equation \(\frac{2}{x} + w = \frac{3}{y}\).To solve for \(y\), we take the reciprocal (invert) both sides of the equation. By inverting, we mean transforming \( a = b \) into \( \frac{1}{a} = \frac{1}{b} \), where both sides of the new equation are reciprocals of the original.Applying this to our equation, \(\frac{1}{\frac{2}{x} + w} = \frac{y}{3}\). It's like flipping the fractions upside down, making it simpler to work with and solve for the target variable.

Algebraic Manipulation

Algebraic manipulation involves various methods and steps to simplify, rearrange, or solve equations. In this exercise, we use several common techniques:

• Isolating the variable
• Inverting fractions
• Multiplying both sides by a constant

After inverting the equation to obtain \(\frac{1}{\frac{2}{x} + w} = \frac{y}{3}\), the next step is to solve for \(y\). We achieve this by multiplying both sides of the equation by \(3\): \(3 \times \frac{1}{\frac{2}{x} + w} = y\). Thus, \(y = \frac{3}{\frac{2}{x} + w}\).By using algebraic manipulation, we systematically solve for the desired variable through a series of logical steps. This approach ensures the solution is correct and the process is clear.