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:

(7x − 2y)/xy = 5  …… (1)

(8x + 7y)/xy = 15  …… (2)

Step 1: Simplify Both Equations

(7x − 2y)/xy = 7/ y − 2/ x

(8x + 7y)/xy = 8/ y + 7/ x

So, equations (1) and (2) become:

7/y − 2/x = 5  …… (3)

8/y + 7/x = 15  …… (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:

7b − 2a = 5  …… (5)

8b + 7a = 15  …… (6)

Step 3: Express One Variable in Terms of the Other

From equation (5):

2a = 7b − 5

⇒ a = ^{7b − 5}/₂  …… (7)

Step 4: Substitute the Value of a in Equation (6)

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

8b + 7( ^{7b − 5}/₂ ) = 15

Multiply both sides by 2:

16b + 49b − 35 = 30

65b = 65

⇒ b = 1

Step 5: Find the Value of a

Substitute b = 1 in equation (7):

a = ^{7(1) − 5}/₂

a = ²/₂

a = 1

Step 6: Find the Values of x and y

Since a = 1/x,

1/x = 1 ⇒ x = 1

Since b = 1/y,

1/y = 1 ⇒ y = 1

Final Answer

∴ The solution of the given system of equations is:

x = 1 and y = 1

Conclusion

Thus, by converting the given equations into linear equations in 1/x and 1/y and using the substitution method, we find that the solution of the given system of equations is (1, 1).