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 − 1) + 1/(y − 2) = 2  …… (1)

6/(x − 1) − 3/(y − 2) = 1  …… (2)

Step 1: Substitute 1/(x − 1) = a and 1/(y − 2) = b

Let 1/(x − 1) = a and 1/(y − 2) = b

Then equations (1) and (2) become:

5a + b = 2  …… (3)

6a − 3b = 1  …… (4)

Step 2: Express One Variable in Terms of the Other

From equation (3):

b = 2 − 5a  …… (5)

Step 3: Substitute the Value of b in Equation (4)

Substitute b from equation (5) into equation (4):

6a − 3(2 − 5a) = 1

6a − 6 + 15a = 1

21a = 7

⇒ a = ¹/₃

Step 4: Find the Value of b

Substitute a = 1/3 in equation (5):

b = 2 − 5(1/3)

b = ¹/₃

Step 5: Find the Values of x and y

Since a = 1/(x − 1),

1/(x − 1) = ¹/₃ ⇒ x − 1 = 3 ⇒ x = 4

Since b = 1/(y − 2),

1/(y − 2) = ¹/₃ ⇒ y − 2 = 3 ⇒ y = 5

Final Answer

∴ The solution of the given system of equations is:

x = 4 and y = 5

Conclusion

Thus, by substituting 1/(x − 1) = a and 1/(y − 2) = b and using the substitution method, we find that the solution of the given system of equations is (4, 5).