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:

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

5/4(x + 2y) − 3/5(3x − 2y) = 61/60  …… (2)

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

Let x + 2y = a and 3x − 2y = b

Then equations (1) and (2) become:

(1/2)a + (5/3)b = −3/2  …… (3)

(5/4)a − (3/5)b = 61/60  …… (4)

Step 2: Remove Fractions

Multiply equation (3) by 6:

3a + 10b = −9  …… (5)

Multiply equation (4) by 60:

75a − 36b = 61  …… (6)

Step 3: Solve the Equations

From equation (5):

3a = −9 − 10b

⇒ a = −3 − 10b/3  …… (7)

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

75( −3 − 10b/3 ) − 36b = 61

−225 − 250b − 36b = 61

−225 − 286b = 61

⇒ −286b = 286

⇒ b = −1

Step 4: Find the Value of a

Substitute b = −1 in equation (7):

a = −3 + 10/3

a = 1/3

Step 5: Find the Values of x and y

We have:

x + 2y = 1/3  …… (8)

3x − 2y = −1  …… (9)

Add equations (8) and (9):

4x = −2/3

⇒ x = −1/6

Substitute x = −1/6 in equation (8):

−1/6 + 2y = 1/3

2y = 1/2

⇒ y = 1/4

Final Answer

∴ The solution of the given system of equations is:

x = −1/6 and y = 1/4

Conclusion

Thus, by substituting x + 2y = a and 3x − 2y = b and using the substitution method, we find that the solution of the given system of equations is (−1/6, 1/4).