Solving the System of Equations: 5/x-1+1/y-2=2 and 6/x-1-3/y-2=1
In this article, we will explore the solution to a system of equations involving fractions:
Equation 1: 5/x - 1 + 1/y - 2 = 2 Equation 2: 6/x - 1 - 3/y - 2 = 1
To solve this system, we need 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) - (3 + 1/y) = 2 Equation 2: (6/x) - (3 + 3/y) = 1
Step 2: Isolate x and y
We can start by isolating x in Equation 1:
(5/x) = 2 + (3 + 1/y) x = 5 / (2 + (3 + 1/y))
Now, substitute this expression for x into Equation 2:
(6 / (5 / (2 + (3 + 1/y)))) - (3 + 3/y) = 1
Step 3: Solve for y
To solve for y, we can multiply both sides of the equation by the least common multiple of the denominators, which is 15y. This will eliminate the fractions:
6(2 + (3 + 1/y)) - 15(3 + 3/y) = 15
Expand and simplify:
12 + 18 + 6/y - 45 - 15/y = 15
Combine like terms:
-15 + 6/y - 15/y = 15 -15 - 9/y = 15
Add 15 to both sides:
-9/y = 30 y = -9/30 y = -3/10
Step 4: Find x
Now that we have found y, we can substitute this value back into one of the original equations to find x. Let's use Equation 1:
(5/x) - (3 + 1/(-3/10)) = 2
Simplify:
(5/x) - (3 - 10/3) = 2 (5/x) - (3 - 10/3) = 2
Combine like terms:
(5/x) - 19/3 = 2 (5/x) = 2 + 19/3 (5/x) = (6 + 19)/3 (5/x) = 25/3
Multiply both sides by x:
5 = (25/3)x
Divide both sides by 25/3:
x = 5 * (3/25) x = 3/5
Solution
The solution to the system of equations is x = 3/5 and y = -3/10.
