Problem 25 - Technical Mathematics with Calculus

Question

Equations with Unknown in Denominator. \(\frac{2}{3 x}+6=5\)

Step-by-Step Solution

Answer

The value of x that solves the equation is \(x=-\frac{2}{3}\).

Step 1: Isolate the term with the unknown in the denominator

To solve the equation \(\frac{2}{3x}+6=5\), start by isolating the term with the unknown in the denominator. This can be done by subtracting 6 from both sides of the equation, which gives us \(\frac{2}{3x}=5-6\).

Step 2: Simplify the equation

Subtract 6 from 5 to simplify the right-hand side of the equation: \(\frac{2}{3x}=-1\).

Step 3: Cross-multiply to remove the denominator

To remove the denominator, cross-multiply by 3x: \(2=-3x\).

Step 4: Solve for the unknown x

To find the value of x, divide both sides by -3: \(x=\frac{2}{-3}\) or \(x=-\frac{2}{3}\).

Key Concepts

Understanding Equations with an Unknown in the Denominator

When solving equations that have variables in the denominator, it's essential to approach the problem with caution. These equations, often called rational equations, can be tricky because division by zero is undefined, thus requiring the variable not to take on values that make the denominator zero.

In the exercise \(\frac{2}{3x}+6=5\), the unknown, \(x\), is beneath the fraction line, which makes it a rational equation. The solution is to manipulate the equation in such a way that the unknown is removed from the denominator. However, you must always remember to check the solution since values that make the original denominator zero must be excluded as possible solutions.

Isolating the Variable in Rational Equations

Isolating the variable is a foundational step in algebra that allows you to focus on the part of the equation that contains the unknown. This is done by performing equivalent operations on both sides of the equation until the term with the variable is by itself.

In our example, subtracting 6 from both sides gives us \(\frac{2}{3x}=5-6\). Simplifying further, we get \(\frac{2}{3x}=-1\), which clearly isolates the variable term on one side of the equation. By understanding and applying the principle of maintaining equal balance (what you do to one side, you must do to the other), we're able to effectively isolate \(x\) for easier calculation.

Using Cross-Multiplication to Solve Rational Equations

Cross-multiplication is a technique used to solve equations involving fractions by eliminating the denominator. It involves multiplying the numerator of one fraction by the denominator of the other fraction, effectively 'crossing over' in a diagonal pattern.

This method comes into play after isolating the variable. In our exercise, once \(\frac{2}{3x}=-1\) is established, cross-multiplication leads to the equation \(2=-3x\). This technique is highly effective as it simplifies the equation to a point where the variable can be isolated further using standard algebraic operations. It's particularly useful with rational equations because it quickly gets rid of the fractions, making the path to finding the value of the unknown much clearer.