Question

The solution of the pair of linear equations: \[ \frac{2x}{3} - \frac{y}{2} = -\frac{1}{6} \quad \text{and} \quad \frac{x}{2} + \frac{2y}{3} = 3 \] is:

• A. x = 2, y = – 3
• B. x = – 2, y = 3
• C. x = 2, y = 3
• D. x = – 2, y = – 3

Answer 1

The Correct Option is C

Step 1: Understand the given system of equations:
We are given the following pair of linear equations: \[ \frac{2x}{3} - \frac{y}{2} = -\frac{1}{6} \quad \text{(1)} \] and \[ \frac{x}{2} + \frac{2y}{3} = 3 \quad \text{(2)} \] We need to solve for \( x \) and \( y \).

Step 2: Eliminate fractions by multiplying each equation by suitable numbers:
To eliminate the fractions, we multiply each equation by the least common multiple (LCM) of the denominators in that equation.
For equation (1), the LCM of 3 and 2 is 6, so multiply both sides of the equation by 6: \[ 6 \times \left( \frac{2x}{3} - \frac{y}{2} \right) = 6 \times \left( -\frac{1}{6} \right) \] This simplifies to: \[ 4x - 3y = -1 \quad \text{(3)} \] For equation (2), the LCM of 2 and 3 is 6, so multiply both sides of the equation by 6: \[ 6 \times \left( \frac{x}{2} + \frac{2y}{3} \right) = 6 \times 3 \] This simplifies to: \[ 3x + 4y = 18 \quad \text{(4)} \]

Step 3: Solve the system of equations:
We now have the system of equations: \[ 4x - 3y = -1 \quad \text{(3)} \] \[ 3x + 4y = 18 \quad \text{(4)} \] We can solve this system using the method of elimination or substitution. Let's use the elimination method. First, multiply equation (3) by 4 and equation (4) by 3 to make the coefficients of \( y \) the same: \[ 4(4x - 3y) = 4(-1) \quad \Rightarrow \quad 16x - 12y = -4 \quad \text{(5)} \] \[ 3(3x + 4y) = 3(18) \quad \Rightarrow \quad 9x + 12y = 54 \quad \text{(6)} \] Now, add equations (5) and (6) to eliminate \( y \): \[ (16x - 12y) + (9x + 12y) = -4 + 54 \] \[ 25x = 50 \] \[ x = \frac{50}{25} = 2 \]

Step 4: Substitute the value of \( x \) into one of the original equations:
Substitute \( x = 2 \) into equation (3): \[ 4(2) - 3y = -1 \] \[ 8 - 3y = -1 \] \[ -3y = -1 - 8 = -9 \] \[ y = \frac{-9}{-3} = 3 \]

Step 5: Conclusion:
The solution to the system of equations is \( x = 2 \) and \( y = 3 \), so the correct answer is \( \boxed{x = 2, y = 3} \).