Solving the System of Linear Equations: 5/x-1+1/y-2=2 and 6/x-1-3/y-2=1
In this article, we will solve a system of linear equations involving fractional expressions. The given equations are:
Equation 1: 5/x - 1 + 1/y - 2 = 2 Equation 2: 6/x - 1 - 3/y - 2 = 1
Our goal is to find the values of x and y that satisfy both equations.
Step 1: Simplify the Equations
Let's simplify each equation by combining like terms:
Equation 1: 5/x - 1/y = 5 Equation 2: 6/x - 3/y = 4
Step 2: Multiply by the Least Common Multiple (LCM)
To eliminate the fractions, we need to multiply each equation by the LCM of the denominators, which is xy.
Equation 1: 5y - x = 5xy Equation 2: 6y - 3x = 4xy
Step 3: Solve the System of Linear Equations
Now we have a system of linear equations in two variables:
Equation 1: 5y - x = 5xy Equation 2: 6y - 3x = 4xy
We can solve this system using substitution or elimination methods. Here, we will use the elimination method.
Step 4: Eliminate x
Multiply Equation 1 by 3 and Equation 2 by 1, then subtract Equation 2 from Equation 1 to eliminate x:
(15y - 3x = 15xy) - (6y - 3x = 4xy) => 9y = 11xy
Step 5: Solve for y
Now, divide both sides by 11x to get:
y = 9/11x
Step 6: Substitute y into One of the Original Equations
Substitute y into Equation 1 to solve for x:
5(9/11x) - x = 5xy 45/11 - x = 5(9/11x)x
Step 7: Solve for x
Simplify and solve for x:
x = 11/14 y = 9/16
Final Answer
The solution to the system of linear equations is x = 11/14 and y = 9/16.
Therefore, the values of x and y that satisfy both equations are x = 11/14 and y = 9/16.
