Matrices & Determinants
System of linear equations with infinitely many solutions
Grade 12
Question:
<p>If the system of equations <math>x + y + z = 5</math>, <math>x + 2y + 3z = 9</math>, <math>x + 3y + az = b</math> has infinitely many solutions, then <math>b - a</math> equals</p>
<p>(a) 8</p>
<p>(b) 18</p>
<p>(c) 21</p>
<p>(d) 5</p>
Step-by-Step Solution
Key Concept: A system of linear equations has infinitely many solutions when all determinants D, D₁, D₂, D₃ equal zero. Use this condition to find the values of the parameters.
<p><strong>Step 1:</strong> Since the system of equations has infinitely many solutions, we have <math>D = D_1 = D_2 = D_3 = 0</math>.</p><p><strong>Step 2:</strong> Calculate <math>D</math>:</p><p><math>D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & a \end{vmatrix} = 1(2a - 9) - 1(a - 3) + 1(-2) = a - 5</math></p><p><strong>Step 3:</strong> From <math>D = 0</math>: <math>a - 5 = 0 \Rightarrow a = 5</math></p><p><strong>Step 4:</strong> Calculate <math>D_3</math>:</p><p><math>D_3 = \begin{vmatrix} 1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 3 & b \end{vmatrix} = 1(2b - 27) - 1(b - 9) + 5(3 - 2) = b - 13</math></p><p><strong>Step 5:</strong> From <math>D_3 = 0</math>: <math>b - 13 = 0 \Rightarrow b = 13</math></p><p><strong>Step 6:</strong> Therefore, <math>b - a = 13 - 5 = 8</math></p><p>∴ Answer is (a).</p>
Correct Answer: A