Problem 25 - Algebra for College Students

Question

Find the solution set to each equation. $$\frac{2}{x}=\frac{3}{4}$$

Step-by-Step Solution

Answer

The solution set to the equation is \ \left\{ \frac{8}{3} \right\} \.

Step 1 - Cross Multiply

To solve the equation \( \frac{2}{x} = \frac{3}{4} \), start by cross multiplying. This means multiplying the numerator of each fraction by the denominator of the other fraction: \ 2 \times 4 = 3 \times x \.

Step 2 - Simplify the Equation

Simplify the result from cross multiplication: \( 2 \times 4 = 8 \ \Rightarrow 8 = 3x \).

Step 3 - Solve for x

Isolate x by dividing both sides of the equation by 3: \( x = \frac{8}{3} \).

Key Concepts

Cross Multiplication

Cross multiplication is a method used to eliminate the fractions in rational equations and convert them into simpler linear equations. It involves multiplying the numerator of one fraction by the denominator of the other fraction. This is especially useful in equations where comparing two ratios or fractions is involved.
For example, in the equation \( \frac{2}{x} = \frac{3}{4} \), you cross multiply by multiplying 2 (numerator of the first fraction) by 4 (denominator of the second fraction) and x (denominator of the first fraction) by 3 (numerator of the second fraction).
Therefore, it becomes:
\[ 2 \times 4 = 3 \times x \]
which simplifies to \( 8 = 3x \). This is the foundation for solving the equation. Cross multiplication helps us deal with rational equations without worrying about the fractions.

Equation Simplification

After cross multiplying, the next step is to simplify the resulting equation. Simplification involves performing arithmetic operations to make the equation more manageable.

• The original cross multiplication yields \( 2 \times 4 = 3 \times x \).
• Perform the multiplication to get \( 8 = 3x \).

Simplifying equations often leads to a linear form which is much easier to solve. The goal is always to bring the equation into an easier state where isolating the variable (like x) is straightforward. Simplification usually involves combining like terms, removing parentheses, or clearing out any fractions that remain.

Isolating Variables

Isolating the variable is the final step in solving the equation. This means getting the variable (e.g., x) alone on one side of the equation.
For example, after simplifying the equation to \( 8 = 3x \), you need to solve for x. To do so:

• You divide both sides of the equation by 3.
• This operation results in \[ x = \frac{8}{3} \].

Isolating the variable ensures we have a clear solution to the equation. It is a fundamental step in algebra where you perform inverse operations to