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 = 5xy  …… (1)

x + 2y = 13xy  …… (2)

Step 1: Divide Both Equations by xy

Divide equation (1) by xy:

x/xy + y/xy = 5

⇒ 1/y + 1/x = 5  …… (3)

Divide equation (2) by xy:

x/xy + 2y/xy = 13

⇒ 1/y + 2/x = 13  …… (4)

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

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

Then equations (3) and (4) become:

a + b = 5  …… (5)

2a + b = 13  …… (6)

Step 3: Solve the Linear System

Subtract equation (5) from equation (6):

(2a + b) − (a + b) = 13 − 5

a = 8

Substitute a = 8 in equation (5):

8 + b = 5

⇒ b = −3

Step 4: Find the Values of x and y

Since a = 1/x,

1/x = 8 ⇒ x = ¹/₈

Since b = 1/y,

1/y = −3 ⇒ y = −¹/₃

Final Answer

∴ The solution of the given system of equations is:

x = ¹/₈ and y = −¹/₃

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 (1/8, −1/3).