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/(3x + y) + 1/(3x − y) = ³/₄  …… (1)

1/2(3x + y) − 1/2(3x − y) = −¹/₈  …… (2)

Step 1: Simplify Equation (2)

1/2(3x + y) − 1/2(3x − y)

= 1/2[(3x + y) − (3x − y)]

= 1/2(2y)

= y

So, equation (2) becomes:

y = −¹/₈  …… (3)

Step 2: Substitute y in Equation (1)

Substitute y = −1/8 in equation (1):

1/(3x − 1/8) + 1/(3x + 1/8) = ³/₄

Step 3: Take LCM of Denominators

LCM = (3x − 1/8)(3x + 1/8)

⇒ (3x + 1/8 + 3x − 1/8) / [(3x − 1/8)(3x + 1/8)] = ³/₄

⇒ 6x / [(3x)² − (1/8)²] = ³/₄

⇒ 6x / (9x² − 1/64) = ³/₄

Step 4: Cross Multiply

4 × 6x = 3(9x² − 1/64)

24x = 27x² − 3/64

Multiply the whole equation by 64:

1536x = 1728x² − 3

⇒ 1728x² − 1536x − 3 = 0

Step 5: Solve the Quadratic Equation

Using quadratic formula:

x = [1536 ± √(1536² + 4 × 1728 × 3)] / (2 × 1728)

x = [1536 ± 1536] / 3456

⇒ x = ¹/₁ or x = ¹/₇₂

Step 6: Find the Corresponding Values of y

Since y = −¹/₈,

For x = 1:

(x, y) = (1, −1/8)

For x = 1/72:

(x, y) = (1/72, −1/8)

Final Answer

∴ The solutions of the given system of equations are:

(1, −¹/₈) and (¹/₇₂, −¹/₈)

Conclusion

Thus, by simplifying the equations and using substitution method, we obtain two solutions of the given system of equations.