Solving the System of Equations using the Elimination Method
In this article, we will solve the system of equations:
Equation 1: 5/x - 1 + 1/y - 2 = 2 Equation 2: 6/x - 1 - 3/y - 2 = 1
using the elimination method.
Step 1: Write the equations
We are given two equations:
5/x - 1 + 1/y - 2 = 2 ... (1) 6/x - 1 - 3/y - 2 = 1 ... (2)
Step 2: Multiply the equations by necessary multiples
We can see that the coefficients of 1/x in both equations are different. To make the coefficients of 1/x the same, we can multiply Equation (1) by 6 and Equation (2) by 5. This will make the coefficients of 1/x equal:
30/x - 6 + 6/y - 12 = 12 ... (3) 30/x - 5 - 15/y - 10 = 5 ... (4)
Step 3: Subtract Equation (4) from Equation (3)
Subtracting Equation (4) from Equation (3), we get:
(30/x - 6 + 6/y - 12) - (30/x - 5 - 15/y - 10) = 12 - 5 -1 + 21/y - 2 = 7
Simplifying the equation, we get:
21/y = 9 y = 21/9 y = 7/3
Step 4: Substitute the value of y in one of the original equations
Substituting the value of y in Equation (1), we get:
5/x - 1 + 1/(7/3) - 2 = 2 5/x - 1 + 3/7 - 2 = 2
Simplifying the equation, we get:
5/x + 3/7 = 5 5/x = 5 - 3/7 5/x = 32/7 x = 5/(32/7) x = 35/32
Step 5: Write the solution
The solution to the system of equations is:
x = 35/32 y = 7/3
Therefore, the values of x and y that satisfy both equations are x = 35/32 and y = 7/3.
