Solving Linear Diophantine Equations for Fractions with Given Denominators
When dealing with mathematical problems involving fractions with specific denominators, the application of linear Diophantine equations can provide a systematic and elegant solution. This article explores the process of finding two fractions whose denominators are 7 and 13, respectively, and whose sum is 33/91, using the concept of linear Diophantine equations. This method not only provides a clear pathway to the solution but also showcases the utility of these equations in solving problems related to fractions.
The Given Problem
We are given two fractions with denominators 7 and 13, respectively, whose sum is equal to 33/91. The fractions can be written as:
x}{7}
Eliminating Fractions
To eliminate the fractions, we multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 91. This yields:
13x - 7y 33
This is a linear Diophantine equation, where we need to find integer solutions for x and y.
Solving the Linear Diophantine Equation
Firstly, we simplify the equation:
13x - 7y 33
To solve this equation, we start by finding the greatest common divisor (GCD) of the coefficients of x and y, which are 13 and 7. The GCD of 13 and 7 is 1, indicating that the equation has infinite integer solutions.
Method 1: Using the Extended Euclidean Algorithm
Let's use the extended Euclidean algorithm to find one particular solution to the equation. We start by expressing 1 as a linear combination of 13 and 7:
13 - 7 6
6 - 7 -1
Combining these, we get:
13 - 2(7) 1
Multiplying both sides by 33, we obtain:
13(33) - 2(7)(33) 33
This gives us one particular solution:
x 33, y -66
However, we can adjust x and y to find a solution where both are positive. We use the general solution of linear Diophantine equations:
x 33 7t
y -66 13t
Method 2: Continued Fraction Method
Another method involves using continued fractions. We start by rewriting the fraction as a mixed fraction:
Continuing the process:
Multiplying both sides by -33, we get:
13 - 33 -1
Multiplying by -33 again, we get:
13(-33) - 7(66) 33
This gives us a particular solution:
x -33, y 66
Using the general solution, we have:
x -33 7t, y 66 - 13t
Finding Positive Solutions
To find positive solutions, we solve the inequalities:
-33 7t > 0
66 - 13t > 0
Solving for t, we get:
t > 33/7 ≈ 4.7
t
The only integer value for t that satisfies both inequalities is t 5. Substituting t 5:
x -33 7(5) 2
y 66 - 13(5) 1
Therefore, the two fractions are:
Conclusion
In conclusion, using linear Diophantine equations and continued fractions, we have successfully determined the fractions with denominators 7 and 13, whose sum is 33/91. The solution highlights the power and flexibility of these mathematical tools in solving complex problems. This process is not only useful for academic purposes but also in a wide range of practical applications, including SEO keyword optimization and content creation.