Solve the Equation: x/4y + y/x - 1 = 0 Using Multiplication Formulas

Consider the following relationship between x and y:

$\frac{x}{4y}+\frac{y}{x}-1=0$

Choose the short multiplication formula that represents this equation.

Video Solution

Solution Steps

00:00 Present the equation as a shortened multiplication formula

00:03 Multiply by the common denominator to eliminate fractions

00:20 Solve the multiplications

00:32 Arrange the equation

00:37 Break down 4 into factors 2 and 2

00:47 Break down 4 into 2 squared

00:50 Now use the shortened multiplication formula and convert to parentheses

00:55 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we'll begin by simplifying the given expression:

The equation is $\frac{x}{4y} + \frac{y}{x} - 1 = 0$ .

Step 1: Eliminate the fraction by obtaining a common denominator for the left side of the equation. The common denominator is $4y \times x = 4xy$ .
Step 2: Rewrite each term with the common denominator:
$\frac{x^2}{4xy} + \frac{4y^2}{4xy} - \frac{4y}{4y} = 0$ . This becomes $\frac{x^2 + 4y^2 - 4xy}{4xy} = 0$ .
Step 3: Multiply through by $4xy$ to eliminate the denominator: $x^2 + 4y^2 - 4xy = 0$ .
Step 4: Recognize this as being equivalent to the expanded form of $(x - 2y)^2 = x^2 - 2xy + 4y^2 = 0$ .
Step 5: Confirm that $x^2 - 2xy + 4y^2 \equiv x^2 + 4y^2 - 4xy$ , indicating $(x - 2y)^2 = 0$ .

Thus, we conclude that the equation can be rewritten using the square of a difference formula:

Therefore, the short multiplication formula for the given equation is $(x - 2y)^2 = 0$ .