Problem 29 - Algebra for College Students

Question

Find the solution set to each equation. $$-\frac{5}{7}=\frac{2}{x}$$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{14}{5} \).

Step 1: Understanding the Equation

The equation given is \(-\frac{5}{7} = \frac{2}{x}\). Our goal is to solve for the variable \(x\).

Step 2: Isolate the Variable

To isolate \(x\), cross-multiply the equation to get: \(-5x = 14\).

Step 3: Solve for \(x\)

Divide both sides by -5 to isolate \(x\): \( x = \frac{14}{-5} = -\frac{14}{5} \).

Step 4: Simplify the Fraction

If necessary, simplify \(-\frac{14}{5}\). In this case, \( -\frac{14}{5} \) is already in its simplest form.

Key Concepts

cross-multiplication

Cross-multiplication is a method used to solve rational equations, where each side of the equation is a fraction.
It helps to eliminate the fractions by multiplying across the diagonal.
This effectively converts a fraction equation into a linear equation.

To use cross-multiplication, follow these steps:

• Identify the two fractions set equal to each other, such as \(\-\frac{5}{7} = \frac{2}{x}\).


• Multiply the numerator of the first fraction by the denominator of the second fraction.


• Do the same for the numerator of the second fraction and the denominator of the first fraction.

In our example, it means multiplying \(-5\) with \(x\) and \(7\) with \(2\), resulting in \-5x = 14.
This step combines both fractions into a single linear equation that can be solved more easily.

isolating variables

Isolating the variable means rearranging the equation so that the variable you're solving for is on one side of the equation by itself.
This process often involves basic algebraic operations like addition, subtraction, multiplication, and division.

Here’s how to isolate the variable in our equation \( -5x = 14 \):

• We have \( -5x = 14 \).


• To isolate \( x \), divide both sides by \( -5 \).


• This gives \( x = \frac{14}{-5} \).

By following these steps, you have effectively isolated \( x \) on one side of the equation and simplified the expression on the other side.

simplifying fractions

Simplifying fractions is the process of making a fraction as simple as possible.
It involves reducing the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

In our problem, we have \( -\frac{14}{5} \):

• First, determine if the numerator and denominator share any common factors.


• The GCD of 14 and 5 is 1, which means the fraction cannot be simplified further.


• Thus, \( -\frac{14}{5} \) is already in its simplest form.

So, the solution to the equation \( -\frac{5}{7} = \frac{2}{x} \) is \( x = -\frac{14}{5} \).
This fraction is simplified and represents the final answer.