Problem 30 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The solution is x = 9.

Step 1: Eliminate Fractions by Finding a Common Denominator

To get rid of the fractions, let's first find a common denominator for the terms on the left-hand side. The denominators are 3 and 2x. A common denominator here would be 6x. Multiply each term by 6x to eliminate the fractions:\(6x \cdot \left( \frac{5}{3} - \frac{3}{2x} \right) = 6x \cdot \frac{3}{2}\).

Step 2: Simplify Each Term

Multiply each term by 6x:- First term: \(6x \cdot \frac{5}{3} = 10x\)- Second term: \(6x \cdot \left(-\frac{3}{2x}\right) = -9\)- Right-hand side: \(6x \cdot \frac{3}{2} = 9x\)So, we have: \(10x - 9 = 9x\).

Step 3: Solve for x

Now, solve for \(x\) by moving terms involving \(x\) to one side of the equation:- Subtract \(9x\) from both sides:\(10x - 9x = 9\).So, \(x = 9\).

Key Concepts

Eliminating Fractions

Eliminating fractions from an equation can make solving much simpler. In the given problem, \( \frac{5}{3} - \frac{3}{2x} = \frac{3}{2} \), we have two fractions on the left side that we want to eliminate. To do that, we need to multiply each term by a number that will make the denominators go away. This process is especially useful when dealing with variables in the denominators, as it turns a complex equation into a simpler one.
For this exercise, the denominators are 3 and 2x, and a smart choice would be to multiply by the least common multiple of these denominators. Choosing a common denominator like 6x helps in clearing the fractions.
After multiplying each term in the equation by 6x, the fractions are eliminated, leading us to a more manageable equation for simplifying further steps. Changing the form of each term in the equation will give us an equation without fractions, making it no different from basic algebraic equations people often solve.

Common Denominator

Finding a common denominator is a key technique for eliminating fractions effectively. When you have more than one fraction in an equation, you can find a number that all denominators can divide into without leaving a remainder.
In our exercise: \(\frac{5}{3} - \frac{3}{2x} = \frac{3}{2}\), the denominators are 3 and 2x. The goal is to find their least common multiple, in this case, it is 6x.
The process involves simply multiplying each term by 6x so that every denominator cancels out.

• Multiplying \(6x\) with \(\frac{5}{3}\) results in \(10x\).
• For \(\frac{3}{2x}\), \(6x\) turns it into \(-9\).
• The right side becomes \(9x\) after multiplying \(\frac{3}{2}\) by \(6x\).

Now, look at the equation without fractions: \(10x - 9 = 9x\). This straightforward expression opens the door to simpler manipulation and solution,made possible by the clever use of a common denominator.

Isolating Variables

Isolating variables involves arranging the equation to get the variable all by itself on one side. This is essential to finding a solution.
After eliminating fractions in our equation with common denominators, we have: \(10x - 9 = 9x\). The goal is to get \(x\) by itself. To achieve this, it's often necessary to move all terms containing \(x\) to one side and numbers to the other.
Here's how this works in practice:

• You subtract \(9x\) from both sides, which results in \(10x - 9x = 9\).
• This simplifies directly to \(x = 9\).

By skillfully moving terms and simplifying, isolating variables transforms the equation into a form that directly shows the value of \(x\).The method is straightforward and a powerful tool in any algebra toolkit. No matter how the equation looks initially, isolating the variable brings you directly to the solution.