Problem 22 - Precalculus

Question

(a) \(\frac{2}{3 x}=\frac{3}{x}\) (b) \(\frac{2}{3 x}=\frac{3}{x+1}\) (c) \(\frac{2}{3 x}=\frac{3}{x}+1\)

Step-by-Step Solution

Answer

(a) \( x = \frac{9}{2} \); (b) \( x = \frac{2}{7} \); (c) \( x = -\frac{7}{3} \).

Step 1: Isolate the Variable (a)

Given the equation \( \frac{2}{3x} = \frac{3}{x} \). Cross-multiply to eliminate the fractions: \( 2x = 3 \times 3x \). This simplifies the equation to \( 2x = 9 \).

Step 2: Solve for x (a)

To find \( x \), divide both sides by 2: \( x = \frac{9}{2} \).

Step 3: Cross-Multiplication (b)

Given \( \frac{2}{3x} = \frac{3}{x+1} \). Use cross-multiplication: \( 2(x+1) = 3 \times 3x \), which simplifies to \( 2x + 2 = 9x \).

Step 4: Rearrange and Solve (b)

Rearrange the equation \( 2x + 2 = 9x \) to \( 2 = 9x - 2x \). Simplify to get \( 2 = 7x \). Then, divide by 7: \( x = \frac{2}{7} \).

Step 5: Initial Rearrangement (c)

Given \( \frac{2}{3x} = \frac{3}{x} + 1 \). Start by subtracting \( \frac{3}{x} \) from both sides: \( \frac{2}{3x} - \frac{3}{x} = 1 \).

Step 6: Find a Common Denominator (c)

To combine the fractions \( \frac{2}{3x} - \frac{3}{x} \), use a common denominator \( 3x \): \( \frac{2 - 9}{3x} = 1 \). Simplify to get \( \frac{-7}{3x} = 1 \).

Step 7: Solve for x (c)

Cross-multiply: \( -7 = 3x \), so \( x = -\frac{7}{3} \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a handy technique used to solve equations involving fractions. When you have an equation like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication allows you to eliminate the fractions by multiplying the numerator of each fraction by the denominator of the other fraction. So, the equation becomes \( a \cdot d = b \cdot c \). This method simplifies the equation to a basic algebraic expression.

Take for example the equation we handled in exercise (a), \( \frac{2}{3x} = \frac{3}{x} \). Applying cross-multiplication, you multiply 2 by \( x \) and 3 by \( 3x \), resulting in the new equation \( 2x = 9 \). This straightforward way of converting a fraction equation into a simple linear equation helps make solving for the variable more intuitive, especially for more complex setups where another direct approach might seem difficult.

Fractions

Fractions represent a part of a whole and are written with two numbers separated by a slash. The number on top is called the numerator, and the number on the bottom is the denominator. Understanding fractions is key to mastering many algebraic techniques.

For instance, when dealing with the equation \( \frac{2}{3x} \), the fraction tells us that 2 is divided by \( 3x \), thereby illustrating how one part relates to the whole \( 3x \). In equation solving, our task often revolves around manipulating these fractions to isolate the variable in question. This might mean performing operations such as cross-multiplication or finding common denominators to make terms easier to work with.

Fractions can sometimes be intimidating, but viewing them as numeric expressions that can be simplified just like whole numbers makes them much simpler to manage.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. Finding a common denominator makes it easier to add, subtract, or even compare fractions.

In our exercise, when faced with \( \frac{2}{3x} - \frac{3}{x} = 1 \), having different denominators can seem challenging. However, a common denominator helps align these so they can be combined easily. Here, the common denominator is \( 3x \), the smallest expression that both original denominators \( 3x \) and \( x \) can divide without leaving a remainder.

Rewriting the fractions with this common denominator allows for simplification, transforming the equations to more manageable forms. For example, \( \frac{2}{3x} - \frac{3}{x} = \frac{2}{3x} - \frac{9}{3x} \) simplifies to \( \frac{-7}{3x} \), making it simple to proceed with solving for the unknown variable afterward.