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:
10/(x + y) + 2/(x − y) = 4 …… (1)
15/(x + y) − 9/(x − y) = −2 …… (2)
Step 1: Substitute (x + y) = a and (x − y) = b
Let x + y = a and x − y = b
Then equations (1) and (2) become:
10/a + 2/b = 4 …… (3)
15/a − 9/b = −2 …… (4)
Step 2: Remove Fractions
Multiply equation (3) by ab:
10b + 2a = 4ab …… (5)
Multiply equation (4) by ab:
15b − 9a = −2ab …… (6)
Step 3: Solve the Equations
From equation (5):
4ab = 10b + 2a …… (7)
Substitute ab from equation (7) into equation (6):
15b − 9a = −2(10b + 2a)/4
Multiply both sides by 4:
60b − 36a = −20b − 4a
80b = 32a
⇒ a = 5/2 b
Step 4: Find the Value of b
Substitute a = 5/2b in equation (5):
10b + 2(5/2b) = 4(5/2b)b
10b + 5b = 10b²
15b = 10b²
⇒ b = 3/2
Step 5: Find the Values of x and y
Now,
x + y = a = 5/2 × 3/2 = 15/4
x − y = b = 3/2
Add both equations:
2x = 15/4 + 3/2
2x = 21/4
⇒ x = 21/8
Substitute x in x − y = 3/2:
21/8 − y = 3/2
⇒ y = 9/8
Final Answer
∴ The solution of the given system of equations is:
x = 21/8 and y = 9/8
Conclusion
Thus, by substituting x + y = a and x − y = b and using the substitution method, we find that the solution of the given system of equations is (21/8, 9/8).