Problem 21 - Algebra Form and Function

Question

If \(f(x)=\frac{x}{2-3 x},\) solve \(f(b)=20\).

Step-by-Step Solution

Answer

Answer: \(b = \frac{40}{61}\)

Step 1: Write down the equation with the function definition

We are given the function \(f(x) = \frac{x}{2-3x}\) and the equation \(f(b) = 20\). Plugging \(f(x)\) into the equation looks like this: $$\frac{b}{2-3b} = 20$$.

Step 2: Simplify the equation

In order to solve for \(b\), we need to get rid of the fraction. To do this, we can multiply both sides of the equation by the denominator, \((2-3b)\): $$(2-3b) \cdot \frac{b}{2-3b} = 20 \cdot (2-3b)$$ This simplifies to: $$b = 20(2-3b)$$

Step 3: Distribute and solve for b

Now we need to distribute the \(20\) on the right side of the equation: $$b = 40 - 60b$$ Next, we'll add \(60b\) to both sides of the equation to isolate \(b\) on the left side: $$61b = 40$$ Finally, we divide both sides by \(61\) to solve for \(b\): $$b = \frac{40}{61}$$

Step 4: Conclusion

The solution to the equation \(f(b) = 20\) for the function \(f(x) = \frac{x}{2-3x}\) is \(b = \frac{40}{61}\).

Key Concepts

Fraction Simplification

When dealing with rational equations such as \( \frac{b}{2-3b} = 20 \), simplifying the fraction is a crucial step. Simplification involves manipulating the equation to eliminate the fraction and make it easier to solve.

• Begin by identifying the fraction in the equation, which here is \( \frac{b}{2-3b} \).
• To remove the fraction, multiply both sides of the equation by the denominator \((2-3b)\).

This operation transforms the equation into one without a fraction: \( b = 20(2-3b) \). By simplifying the equation this way, you make the next steps of solving for the variable more straightforward. It's important to ensure that the denominator is not zero at any point during the simplification, so always check your algebraic manipulations.

Isolation of Variable

Isolating the variable is vital in solving equations because it allows us to express the variable explicitly. In this exercise, we aim to isolate \( b \) in the equation \( b = 20(2-3b) \).Here's how you do it:

• Start by distributing the multiplication on the right side of the equation: \( b = 40 - 60b \).
• The next step is to bring all terms involving the variable to one side of the equation and constants to the other.
• Add \( 60b \) to both sides to keep the variable on the left: \( 61b = 40 \).
• Finally, divide both sides by \( 61 \) to isolate \( b \): \( b = \frac{40}{61} \).

Through these steps, we effectively isolate \( b \), achieving a simple expression to determine its value. Checking each step will help verify that the equation is manipulated correctly.

Distribution in Equations

Distribution is an essential operation when solving equations that involve expressions like \( 20(2-3b) \). This involves multiplying each term inside the parenthesis by the factor outside.

• In this problem, start with: \( b = 20(2-3b) \).
• Distribute \( 20 \) across the terms inside the parenthesis: \( b = 20 \times 2 - 20 \times 3b \).
• This simplifies to \( b = 40 - 60b \).

Distribution helps break down components of an equation into simpler parts, making it easier to isolate and solve for the variable. In every step, carefully multiply and simplify to avoid errors. This ensures that subsequent operations proceed smoothly, leading to the correct solution.