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:

2(3u − v) = 5uv  …… (1)

2(u + 3v) = 5uv  …… (2)

Step 1: Divide Both Equations by uv

Divide equation (1) by uv:

2(3u − v)/uv = 5

⇒ 6/v − 2/u = 5  …… (3)

Divide equation (2) by uv:

2(u + 3v)/uv = 5

⇒ 2/v + 6/u = 5  …… (4)

Step 2: Substitute 1/u = x and 1/v = y

Let 1/u = x and 1/v = y

Then equations (3) and (4) become:

6y − 2x = 5  …… (5)

2y + 6x = 5  …… (6)

Step 3: Solve the Linear System

From equation (5):

6y = 5 + 2x

⇒ y = ^{5 + 2x}/₆  …… (7)

Substitute y from equation (7) into equation (6):

2( ^{5 + 2x}/₆ ) + 6x = 5

^{5 + 2x}/₃ + 6x = 5

Multiply the whole equation by 3:

5 + 2x + 18x = 15

20x = 10

⇒ x = ¹/₂

Step 4: Find the Value of y

Substitute x = 1/2 in equation (7):

y = ^{5 + 2(1/2)}/₆

y = ⁶/₆

y = 1

Step 5: Find the Values of u and v

Since x = 1/u,

1/u = ¹/₂ ⇒ u = 2

Since y = 1/v,

1/v = 1 ⇒ v = 1

Final Answer

∴ The solution of the given system of equations is:

u = 2 and v = 1

Conclusion

Thus, by dividing the equations by uv and using the substitution method, we find that the solution of the given system of equations is (u, v) = (2, 1).