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:

3/(x + y) + 2/(x − y) = 2  …… (1)

9/(x + y) − 4/(x − y) = 1  …… (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:

3/a + 2/b = 2  …… (3)

9/a − 4/b = 1  …… (4)

Step 2: Remove Fractions

Multiply equation (3) by ab:

3b + 2a = 2ab  …… (5)

Multiply equation (4) by ab:

9b − 4a = ab  …… (6)

Step 3: Solve the Equations

From equation (6):

ab = 9b − 4a  …… (7)

Substitute ab from equation (7) into equation (5):

3b + 2a = 2(9b − 4a)

3b + 2a = 18b − 8a

10a = 15b

⇒ a = ³/₂ b

Step 4: Find the Value of b

Substitute a = ³/₂b in equation (7):

ab = 9b − 4a

(³/₂b) b = 9b − 4(³/₂b)

³/₂b² = 9b − 6b

³/₂b² = 3b

⇒ b = 2

Step 5: Find the Values of x and y

Now,

x + y = a = ³/₂ × 2 = 3

x − y = b = 2

Add both equations:

2x = 5

⇒ x = ⁵/₂

Substitute x in x − y = 2:

⁵/₂ − y = 2

⇒ y = ¹/₂

Final Answer

∴ The solution of the given system of equations is:

x = ⁵/₂ and y = ¹/₂

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 (5/2, 1/2).