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, where x ≠ −1 and y ≠ 1:

5/(x + 1) − 2/(y − 1) = ¹/₂  …… (1)

10/(x + 1) + 2/(y − 1) = ⁵/₂  …… (2)

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

Let 1/(x + 1) = a and 1/(y − 1) = b

Then equations (1) and (2) become:

5a − 2b = ¹/₂  …… (3)

10a + 2b = ⁵/₂  …… (4)

Step 2: Remove Fractions

Multiply equations (3) and (4) by 2:

10a − 4b = 1  …… (5)

20a + 4b = 5  …… (6)

Step 3: Solve the Linear System

Add equations (5) and (6):

30a = 6

⇒ a = ¹/₅

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

10(1/5) − 4b = 1

2 − 4b = 1

⇒ 4b = 1

⇒ b = ¹/₄

Step 4: Find the Values of x and y

Since a = 1/(x + 1),

1/(x + 1) = ¹/₅ ⇒ x + 1 = 5 ⇒ x = 4

Since b = 1/(y − 1),

1/(y − 1) = ¹/₄ ⇒ y − 1 = 4 ⇒ 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 − 1) = b and using the substitution method, we find that the solution of the given system of equations is (4, 5).