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 ≠ 0 and y ≠ 0:

x + y = 2xy  …… (1)

(x − y)/xy = 6  …… (2)

Step 1: Divide Equation (1) by xy

Divide equation (1) by xy:

x/xy + y/xy = 2

⇒ 1/y + 1/x = 2  …… (3)

Step 2: Simplify Equation (2)

(x − y)/xy = x/xy − y/xy

⇒ 1/y − 1/x = 6  …… (4)

Step 3: Substitute 1/x = a and 1/y = b

Let 1/x = a and 1/y = b

Then equations (3) and (4) become:

a + b = 2  …… (5)

b − a = 6  …… (6)

Step 4: Solve the Linear System

Add equations (5) and (6):

(a + b) + (b − a) = 2 + 6

2b = 8

⇒ b = 4

Substitute b = 4 in equation (5):

a + 4 = 2

⇒ a = −2

Step 5: Find the Values of x and y

Since a = 1/x,

1/x = −2 ⇒ x = −¹/₂

Since b = 1/y,

1/y = 4 ⇒ y = ¹/₄

Final Answer

∴ The solution of the given system of equations is:

x = −¹/₂ and y = ¹/₄

Conclusion

Thus, by dividing the equations by xy and using the substitution method, we find that the solution of the given system of equations is (−1/2, 1/4).