Problem 2 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank. To clear the equation \(\frac{4}{x}+5=\frac{6}{x-3}\) of fractions, multiply each side of the equation by the least common denominator _____ .

Step-by-Step Solution

Answer

The least common denominator to clear the equation is \(x(x-3)\).

Step 1: Identifying the denominators

The first task is to identify the denominators present in the equation. The equation given is \(\frac{4}{x}+5=\frac{6}{x-3}\). Here, the denominators are 'x' and 'x-3'.

Step 2: Calculating the Least Common Denominator

As the denominators are 'x' and 'x-3', they are two different linear factors and not multiples of each other. So, their least common multiple (LCM) or least common denominator (LCD) would be obtained by their multiplication, which is \(x * (x-3)\) or \(x(x-3)\).

Step 3: Final Answer

So, to clear the equation of fractions, each side of the equation needs to be multiplied by the LCD, which is \(x(x-3)\).

Key Concepts

Least Common Denominator

When dealing with equations that contain fractions, it is important to have a strategy for simplifying them. One helpful method is to use the Least Common Denominator (LCD). The least common denominator of a set of fractions is the smallest expression or number that all the denominators in the equation can divide into without a remainder.

Finding the LCD involves:

• Identifying all the different denominators present in the equation.
• Determining the least common multiple (LCM) of these denominators.
• If the denominators include variables, the LCD usually consists of multiplying these distinct factors together.

The goal is to eliminate the fractions by creating a common baseline, simplifying the equation-solving process. For example, in the equation \(\frac{4}{x}+5=\frac{6}{x-3}\), the denominators are \(x\) and \(x-3\). Their LCD is the product \(x(x-3)\). This LCD allows for clearing the fractions when used to multiply all terms of the equation.

Clearing Fractions

Clearing fractions from an equation is a clever trick to simplify and streamline the solving process. By removing fractions, you transform a complex equation into something more manageable, often making it easier to see the solution.

Here’s how to clear fractions effectively:

• First, determine the least common denominator (LCD) of all fractions involved, as discussed earlier.
• Multiply every term in the equation by the LCD. This uniformly distributes the LCD, clearing the fractions from each term.
• The multiplication transforms each fraction into a whole number or a simpler expression.

For instance, in the equation \(\frac{4}{x}+5=\frac{6}{x-3}\), multiplying by the LCD \(x(x-3)\) results in a "cleared" equation without fractions. This step greatly aids in straightforward solving by converting complex rational expressions into simple linear terms.

Equation Solving

Once fractions are cleared from the equation, solving can proceed using more straightforward algebraic techniques. These include various strategies depending on the type of equation.

Here are some common methods for solving simple equations post-fraction clearing:

• Combine like terms: Collect and simplify terms on both sides of the equation that are similar.
• Isolation of Variables: Rearrange the equation to get the variable of interest "alone" on one side of the equation, often via addition or subtraction.
• Equating both sides: Utilize balance techniques such as dividing or multiplying all terms to isolate or evaluate the variable.

Using the example equation \(\frac{4}{x}+5=\frac{6}{x-3}\), after clearing fractions, you would likely find a simpler equation that could be solved by these techniques.

Each step brings you closer to finding the solution to the equation, ensuring a logical path through algebraic principles.