Problem 60 - Algebra for College Students

Question

Solve each equation. $$\frac{-3}{2 x}=\frac{1}{-5}$$

Step-by-Step Solution

Answer

x = 7.5

Step 1 - Cross-Multiply

To solve the equation \(\frac{-3}{2x} = \frac{1}{-5}\), start by cross-multiplying the fractions. This means multiplying the numerator of each fraction by the denominator of the other fraction. Thus, you get: \-3 \times -5 = 1 \times 2x\Which simplifies to: \15 = 2x.\

Step 2 - Isolate X

To solve for \(x\), divide both sides of the equation by 2:\15 = 2x\x = \frac{15}{2}\.\

Step 3 - Simplify the Solution

Simplify the fraction if necessary: \(x = 7.5\). Now, the solution to the equation is \(x = 7.5\).

Key Concepts

Cross Multiplication

When solving rational equations, one of the key techniques is cross multiplication. This involves multiplying the numerator of one fraction by the denominator of the other to eliminate the fractions. In the equation \(\frac{-3}{2x} = \frac{1}{-5}\), you start by cross-multiplying:

• Multiply -3 by -5 which gives you 15.


• Then, multiply 1 by 2x, providing 2x.

This results in the equation \(15 = 2x\). By cross-multiplying, you're converting an equation with fractions into a simpler form without fractions, making it easier to solve.

Isolating Variable

The next step in solving the equation is to isolate the variable, usually represented as 'x.' To do this, you perform algebraic operations that move all terms involving 'x' to one side of the equation and all constants to the other. After cross-multiplying and obtaining \(15 = 2x\), you need to isolate x by dividing both sides of the equation by 2:

• So, \(x = \frac{15}{2}\).

This step is crucial because it gives you the value of 'x' directly. Remember, the goal of isolating the variable is to have 'x' alone on one side of the equation.

Fraction Simplification

The final step in solving rational equations usually involves simplifying the fractions. In our example, after isolating the variable, we found \(x = \frac{15}{2}\). Simplifying this fraction, if possible, makes the solution easier to understand and more readable.

• Since 15 divided by 2 equals 7.5, the simplified solution is \(x = 7.5\).
  
• Even though \(\frac{15}{2}\) is already simplified in terms of prime factors, it's always a good practice to convert fractions to their simplest form unless specified otherwise.

Fraction simplification helps in verifying the solution and ensuring it's in its most understandable form, which is especially useful for checking accuracy in your final answer.