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:

22/(x + y) + 15/(x − y) = 5  …… (1)

55/(x + y) + 45/(x − y) = 14  …… (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:

22/a + 15/b = 5  …… (3)

55/a + 45/b = 14  …… (4)

Step 2: Remove Fractions

Multiply equation (3) by ab:

22b + 15a = 5ab  …… (5)

Multiply equation (4) by ab:

55b + 45a = 14ab  …… (6)

Step 3: Express One Variable in Terms of the Other

From equation (5):

15a = 5ab − 22b

⇒ a = ^{5ab − 22b}/₁₅

Step 4: Substitute the Value of a in Equation (6)

Substitute a from equation (5) and simplify by elimination:

Multiply equation (5) by 3:

45a + 66b = 15ab  …… (7)

Subtract equation (6) from equation (7):

(45a + 66b) − (45a + 55b) = 15ab − 14ab

11b = ab

⇒ a = 11

Step 5: Find the Value of b

Substitute a = 11 in equation (5):

22b + 15(11) = 5(11)b

22b + 165 = 55b

33b = 165

⇒ b = 5

Step 6: Find the Values of x and y

We have:

x + y = 11  …… (8)

x − y = 5  …… (9)

Add equations (8) and (9):

2x = 16

⇒ x = 8

Substitute x = 8 in equation (9):

8 − y = 5

⇒ y = 3

Final Answer

∴ The solution of the given system of equations is:

x = 8 and y = 3

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 (8, 3).