Solve the System of Linear 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:

2/√x − 3/√y = 2  …… (1)

4/√x − 9/√y = −1  …… (2)

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

Let 1/√x = a and 1/√y = b

Then equations (1) and (2) become:

2a − 3b = 2  …… (3)

4a − 9b = −1  …… (4)

Step 2: Express One Variable in Terms of the Other

From equation (3):

2a − 3b = 2

⇒ 2a = 2 + 3b

⇒ a = ^{2 + 3b}/₂  …… (5)

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

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

4( ^{2 + 3b}/₂ ) − 9b = −1

2(2 + 3b) − 9b = −1

4 + 6b − 9b = −1

−3b = −5

⇒ b = ⁵/₃

Step 4: Find the Value of a

Substitute b = 5/3 in equation (5):

a = ^{2 + 3(5/3)}/₂

a = ^{2 + 5}/₂

a = ⁷/₂

Step 5: Find the Values of x and y

Since a = 1/√x,

1/√x = 7/2 ⇒ √x = 2/7 ⇒ x = ⁴/₄₉

Since b = 1/√y,

1/√y = 5/3 ⇒ √y = 3/5 ⇒ y = ⁹/₂₅

Final Answer

∴ The solution of the given system of equations is:

x = ⁴/₄₉ and y = ⁹/₂₅

Conclusion

Thus, by substituting 1/√x = a and 1/√y = b and using the substitution method, we find that the solution of the given system of equations is (4/49, 9/25).