Solve the System of Equations Using Substitution Method
Video Explanation
Watch the video below to understand the complete solution step by step:
Solution
Question: Solve the following system of equations, where x + y ≠ 0 and x − y ≠ 0:
xy/(x + y) = 6/5 …… (1)
xy/(y − x) = 6 …… (2)
Step 1: Cross Multiply Both Equations
From equation (1):
5xy = 6(x + y)
⇒ 5xy − 6x − 6y = 0 …… (3)
From equation (2):
xy = 6(y − x)
⇒ xy − 6y + 6x = 0 …… (4)
Step 2: Divide Equation (3) by xy
Divide equation (3) by xy:
5 − 6/y − 6/x = 0
⇒ 1/x + 1/y = 5/6 …… (5)
Step 3: Divide Equation (4) by xy
Divide equation (4) by xy:
1 − 6/x + 6/y = 0
⇒ 1/x − 1/y = −1/6 …… (6)
Step 4: Substitute 1/x = a and 1/y = b
Let 1/x = a and 1/y = b
Then equations (5) and (6) become:
a + b = 5/6 …… (7)
a − b = −1/6 …… (8)
Step 5: Solve the Linear System
Add equations (7) and (8):
2a = 4/6
⇒ a = 1/3
Substitute a in equation (7):
1/3 + b = 5/6
⇒ b = 1/2
Step 6: Find the Values of x and y
Since a = 1/x,
1/x = 1/3 ⇒ x = 3
Since b = 1/y,
1/y = 1/2 ⇒ y = 2
Final Answer
∴ The solution of the given system of equations is:
x = 3 and y = 2
Conclusion
Thus, by simplifying the given equations and using the substitution method, we find that the solution of the given system of equations is (3, 2).