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, where x + y ≠ 0 and x − y ≠ 0:

6/(x + y) = 7/(x − y) + 3  …… (1)

1/2(x + y) = 1/3(x − y)  …… (2)

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

Let x + y = a and x − y = b

Then equations (1) and (2) become:

6/a = 7/b + 3  …… (3)

1/(2a) = 1/(3b)  …… (4)

Step 2: Simplify Equation (4)

From equation (4):

1/(2a) = 1/(3b)

Cross multiply:

3b = 2a

⇒ b = ^2a/₃  …… (5)

Step 3: Substitute the Value of b in Equation (3)

Substitute b from equation (5) into equation (3):

6/a = 7/(2a/3) + 3

6/a = ²¹/_2a + 3

Convert 3 into fraction with denominator 2a:

6/a = ^{21 + 6a}/_2a

Multiply both sides by 2a:

12 = 21 + 6a

6a = −9

⇒ a = −³/₂

Step 4: Find the Value of b

Substitute a = −3/2 in equation (5):

b = ^2(−3/2)/₃

b = −1

Step 5: Find the Values of x and y

We have:

x + y = −³/₂  …… (6)

x − y = −1  …… (7)

Add equations (6) and (7):

2x = −³/₂ − 1

2x = −⁵/₂

⇒ x = −⁵/₄

Substitute x in equation (7):

−⁵/₄ − y = −1

y = −¹/₄

Final Answer

∴ The solution of the given system of equations is:

x = −⁵/₄ and y = −¹/₄

Conclusion

Thus, by substituting x + y = a and x − y = b and using the substitution method, we find that the solution of the given system of equations is (−5/4, −1/4).