Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation using two methods. Then explain which method you prefer. $$\frac{3 x}{8}-\frac{4 x}{3}=4$$

Step-by-Step Solution

Answer

The solution to the equation is x equals \(\frac{96}{23}\). Both methods arrived at the same answer, but with less fractions involved, the second method might be simpler and hence preferable.

Step 1: Method 1: Isolating x

In order to isolate the variable, x, add \(\frac{4x}{3}\) to both sides to get: \(\frac{3x}{8} = 4 + \(\frac{4x}{3}\). To further isolate x, multiply out the denominators (common denominator is 24): \(9x = 96 + 32x\), and further simplify to get \(23x = 96\). Finally, solve for x by dividing both sides by 23 to get \(x = \(\frac{96}{23}\))

Step 2: Method 2: Clearing fractions

In order to clear the fractions, multiply each term of the equation by the least common multiple (LCM) of 8 and 3 which is 24: \(9x - 32x = 96\). Rearrange and solve for x to get \(23x = 96\). Solve for x by dividing both sides by 23 to get \(x = \(\frac{96}{23}\))

Step 3: Preference Explanation

Both of these methods arrive at the same solution. However, clearing fractions (Method 2) might appear a little more intuitive and straightforward by getting rid of fractions right at the beginning, which often simplifies the calculation and reduces errors.

Key Concepts

Isolating Variables

Isolating variables is one of the fundamental skills needed to solve linear equations effectively. In this process, the goal is to rearrange the equation such that the variable we want to solve for stands alone on one side of the equation, while all other terms are moved to the opposite side. This is typically done using basic algebraic operations: addition, subtraction, multiplication, and division.

For example, in the given equation \(\frac{3 x}{8}-\frac{4 x}{3}=4\), we begin by adding \(\frac{4x}{3}\) to both sides to get our variable \(x\) on one side of the equation. This demonstrates the first step in isolating the variable. We continue by dealing with the denominators to further isolate \(x\), leading us to multiply each term by a common denominator to eliminate fractions. The final step to isolate the variable is to divide by the coefficient that stands before the variable, leaving us with \(x = \frac{96}{23}\). This step-by-step approach simplifies the equation to make the solution clear.

Clearing Fractions in Equations

Clearing fractions from equations can make solving them less intimidating and mor straightforward. The idea is to eliminate the fractions as one of the first steps by finding a common multiple that each denominator can be multiplied by to produce whole numbers.

To clear fractions in our example equation \(\frac{3 x}{8}-\frac{4 x}{3}=4\), we first identify the least common multiple (LCM) of the denominators, which in this case, are 8 and 3. The LCM is 24. We then multiply each term by 24, the LCM, to eliminate the fractions entirely. This results in a new equation without fractions: \(9x - 32x = 96\), which is significantly easier to solve and rearrange, leading us to \(x = \frac{96}{23}\). Clearing fractions early on helps to streamline the problem-solving process and can minimize errors due to complex fraction operations.

Least Common Multiple (LCM)

Understanding the least common multiple (LCM) is not only a core mathematical concept but also an immensely practical tool when solving equations with fractions. The LCM of two or more numbers is the smallest number that is a multiple of all the numbers involved.

In the context of clearing fractions in equations, the LCM is used to find a common denominator to which all terms can be converted, thereby eliminating the fractions. The LCM for 8 and 3, for example, is 24, because 24 is the smallest number that both 8 and 3 can divide into evenly. Multiplying through by the LCM makes every term in the equation a whole number, which simplifies the equation solving process. Recognizing and applying the LCM can save time and reduce the likelihood of making mistakes when working with fractions in equations.