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

Step 2: Eliminate the fractions:
To simplify the equations, let's eliminate the denominators by multiplying both sides of each equation by the least common multiple (LCM) of the denominators. For the first equation, the LCM of 3 and 2 is 6. Multiply the entire equation by 6:
\[ 6 \times \left( \frac{2x}{3} - \frac{y}{2} \right) = 6 \times \left( -\frac{1}{6} \right) \] \[ 4x - 3y = -1 \quad \text{(Equation 1)} \] For the second equation, the LCM of 2 and 3 is 6. Multiply the entire equation by 6:
\[ 6 \times \left( \frac{x}{2} + \frac{2y}{3} \right) = 6 \times 3 \] \[ 3x + 4y = 18 \quad \text{(Equation 2)} \]

Step 3: Solve the system of equations:
We now have the system of equations:
\[ 4x - 3y = -1 \quad \text{(Equation 1)} \] \[ 3x + 4y = 18 \quad \text{(Equation 2)} \] We can solve this system using the method of elimination or substitution. Let's use the elimination method.

Multiply Equation 1 by 4 and Equation 2 by 3 to make the coefficients of \( y \) in both equations equal:
\[ 4 \times (4x - 3y) = 4 \times (-1) \quad \Rightarrow \quad 16x - 12y = -4 \quad \text{(Equation 3)} \] \[ 3 \times (3x + 4y) = 3 \times 18 \quad \Rightarrow \quad 9x + 12y = 54 \quad \text{(Equation 4)} \] Now add Equation 3 and Equation 4:
\[ (16x - 12y) + (9x + 12y) = -4 + 54 \] \[ 25x = 50 \] \[ x = \frac{50}{25} = 2 \]

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

Conclusion:
The solution of the system of equations is \( x = 2 \) and \( y = 3 \).