Proving the Relationship Between x/5 and xy/8

In this article, we will explore and prove the relationship between the expressions x/5 and xy/8. The proof will involve a series of algebraic manipulations and substitutions, ultimately validating the equation:

x/5 xy/8if and only if x/5 y/3.

Step-by-Step Proof

Given the equation:

x/5 y/3

We need to prove that:

x/5 xy/8

Step 1: Cross-Multiplication

Start with the given equation:

x/5 y/3

Cross multiply to eliminate the fractions:

3x 5y

Step 2: Express y in Terms of x

From the equation 3x 5y, we can express y as:

y (3/5)x

Step 3: Substitute y into xy/8

Now, substitute y (3/5)x into the expression xy/8:

xy/8 x(3/5)x/8 (8/5)x/8 x/5

Conclusion

Therefore, we have shown that:

x/5 xy/8if and only if x/5 y/3

Proof by Backward Manipulation

Let's verify the relationship by starting from the bottom line and working our way up:

Starting with:

x/5 xy/8

Assume x/5 y/3, then we can substitute the value of x from the given condition:

x (5/3)y

Substituting x (5/3)y into xy/8:

xy/8 ((5/3)y)y/8 (8/3)y/8 y/3

This confirms that: