Question

If the system of equations \( x + 5y + 6z = 4 \), \( 2x + 2y + 4z = 1 \) and \( x + y + az = b \) has infinite numbers of solutions then point \( (a,b) \) lies on-

• A. \( x - 2y = 1 \)
• B. \( x - y = 3 \)
• C. \( x + y = 2 \)
• D. \( y - x = 3 \)

Hint:

For infinitely many solutions in a system of three linear equations, one equation must be obtainable as a linear combination of the other two. Compare coefficients carefully to find the required parameters.

Answer 1

The Correct Option is A

Step 1: Write the given system of equations.
The three equations are: \[ x + 5y + 6z = 4, \] \[ 2x + 2y + 4z = 1, \] \[ x + y + az = b. \] For the system to have infinitely many solutions, the third equation must be a linear combination of the first two equations.
Step 2: Express the third equation as a linear combination.
Assume that the third equation can be obtained by multiplying the first equation by \( \lambda \) and the second equation by \( \mu \), and then adding them: \[ \lambda(x + 5y + 6z = 4) + \mu(2x + 2y + 4z = 1). \] This combination should produce: \[ x + y + az = b. \] Comparing coefficients of \( x \) and \( y \), we obtain: \[ \lambda + 2\mu = 1, \] \[ 5\lambda + 2\mu = 1. \] Subtracting the first equation from the second: \[ 4\lambda = 0, \] \[ \lambda = 0. \] Substituting this value into the equation \( \lambda + 2\mu = 1 \): \[ 2\mu = 1 \quad \Rightarrow \quad \mu = \frac{1}{2}. \]
Step 3: Determine the values of \( a \) and \( b \).
Now compare the coefficient of \( z \): \[ a = 6\lambda + 4\mu = 6(0) + 4\left(\frac{1}{2}\right) = 2. \] Next, compare the constant terms: \[ b = 4\lambda + \mu = 4(0) + \frac{1}{2} = \frac{1}{2}. \] Thus, the point is: \[ (a, b) = \left(2, \frac{1}{2}\right). \]
Step 4: Identify the equation satisfied by \( (a, b) \).
Substitute \( x = 2 \) and \( y = \frac{1}{2} \) into the given options. For the equation \( x - 2y = 1 \): \[ 2 - 2\left(\frac{1}{2}\right) = 2 - 1 = 1, \] which satisfies the equation. Conclusion:
Therefore, the point \( (a, b) = \left(2, \frac{1}{2}\right) \) lies on the line:
Final Answer: \( x - 2y = 1 \).