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/(3x + 2y) + 3/(3x − 2y) = ¹⁷/₅  …… (1)

5/(3x + 2y) + 1/(3x − 2y) = 2  …… (2)

Step 1: Substitute (3x + 2y) = a and (3x − 2y) = b

Let 3x + 2y = a and 3x − 2y = b

Then equations (1) and (2) become:

2/a + 3/b = ¹⁷/₅  …… (3)

5/a + 1/b = 2  …… (4)

Step 2: Substitute 1/a = u and 1/b = v

Let 1/a = u and 1/b = v

Then equations (3) and (4) become:

2u + 3v = ¹⁷/₅  …… (5)

5u + v = 2  …… (6)

Step 3: Solve the Linear System

From equation (6):

v = 2 − 5u  …… (7)

Substitute v from equation (7) into equation (5):

2u + 3(2 − 5u) = ¹⁷/₅

2u + 6 − 15u = ¹⁷/₅

−13u + 6 = ¹⁷/₅

Convert 6 into fraction:

−13u + ³⁰/₅ = ¹⁷/₅

−13u = −¹³/₅

⇒ u = ¹/₅

Substitute u = 1/5 in equation (7):

v = 2 − 5(1/5)

v = 1

Step 4: Find the Values of a and b

Since u = 1/a,

1/a = ¹/₅ ⇒ a = 5

Since v = 1/b,

1/b = 1 ⇒ b = 1

Step 5: Find the Values of x and y

We have:

3x + 2y = 5  …… (8)

3x − 2y = 1  …… (9)

Add equations (8) and (9):

6x = 6

⇒ x = 1

Substitute x = 1 in equation (8):

3(1) + 2y = 5

2y = 2

⇒ y = 1

Final Answer

∴ The solution of the given system of equations is:

x = 1 and y = 1

Conclusion

Thus, by substituting 3x + 2y = a and 3x − 2y = b and using the substitution method, we find that the solution of the given system of equations is (1, 1).