<p>If the system of equations</p><p>x + y + z = 6</p><p>2x + 5y + az = b</p><p>x + 2y + 3z = 14</p><p>has infinitely many solutions, then a + b is equal to :</p>
Step-by-Step Solution
Key Concept: For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and the augmented matrix must have a rank less than the number of variables, implying the planes are dependent.
<p>The system is:</p><p>x + y + z = 6</p><p>2x + 5y + az = b</p><p>x + 2y + 3z = 14</p><p>For infinitely many solutions, the determinant of the coefficient matrix must be 0:</p><p>| 1 1 1 |</p><p>| 2 5 a | = 0</p><p>| 1 2 3 |</p><p>1(15 - 2a) - 1(6 - a) + 1(4 - 5) = 0</p><p>15 - 2a - 6 + a - 1 = 0</p><p>8 - a = 0 => a = 8</p><p>Now substitute a = 8 into the system:</p><p>x + y + z = 6 (i)</p><p>2x + 5y + 8z = b (ii)</p><p>x + 2y + 3z = 14 (iii)</p><p>Subtract (i) from (iii): y + 2z = 8 => y = 8 - 2z</p><p>Substitute y into (i): x + 8 - 2z + z = 6 => x = z - 2</p><p>Substitute x and y into (ii): 2(z - 2) + 5(8 - 2z) + 8z = b</p><p>2z - 4 + 40 - 10z + 8z = b</p><p>36 = b</p><p>Thus, a + b = 8 + 36 = 44.</p>
Correct Answer: (3)