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:

5/(x + y) − 2/(x − y) = −1  …… (1)

15/(x + y) + 7/(x − y) = 10  …… (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:

5/a − 2/b = −1  …… (3)

15/a + 7/b = 10  …… (4)

Step 2: Remove Fractions

Multiply equation (3) by ab:

5b − 2a = −ab  …… (5)

Multiply equation (4) by ab:

15b + 7a = 10ab  …… (6)

Step 3: Solve the Equations

From equation (5):

5b − 2a = −ab

⇒ ab = 2a − 5b  …… (7)

Substitute ab from equation (7) into equation (6):

15b + 7a = 10(2a − 5b)

15b + 7a = 20a − 50b

65b = 13a

⇒ a = 5b

Step 4: Find the Value of b

Substitute a = 5b in equation (7):

ab = 2a − 5b

5b × b = 2(5b) − 5b

5b² = 10b − 5b

5b² = 5b

⇒ b = 1

Step 5: Find the Values of x and y

Now,

x + y = a = 5

x − y = b = 1

Add both equations:

2x = 6

⇒ x = 3

Substitute x = 3 in x − y = 1:

3 − y = 1

⇒ y = 2

Final Answer

∴ The solution of the given system of equations is:

x = 3 and y = 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 (3, 2).