Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p><strong>61.</strong> If the system of equations<br>\(x + y + z = 5\)<br>\(x + 2y + 3z = 9\)<br>\(x + 3y + \alpha z = \beta\)<br>has infinitely many solutions, then \(\beta - \alpha\) equals ______.</p>
Step-by-Step Solution
Key Concept: For infinitely many solutions, the system must be dependent (rank of coefficient matrix = rank of augmented matrix < 3). The third equation must be a linear combination of the first two equations.
<p><strong>Step 1:</strong> Write the augmented matrix:</p><p>$$\begin{bmatrix} 1 & 1 & 1 & | & 5 \\ 1 & 2 & 3 & | & 9 \\ 1 & 3 & \alpha & | & \beta \end{bmatrix}$$</p><p><strong>Step 2:</strong> Perform row operations: R₂ → R₂ - R₁ and R₃ → R₃ - R₁:</p><p>$$\begin{bmatrix} 1 & 1 & 1 & | & 5 \\ 0 & 1 & 2 & | & 4 \\ 0 & 2 & \alpha-1 & | & \beta-5 \end{bmatrix}$$</p><p><strong>Step 3:</strong> Perform R₃ → R₃ - 2R₂:</p><p>$$\begin{bmatrix} 1 & 1 & 1 & | & 5 \\ 0 & 1 & 2 & | & 4 \\ 0 & 0 & \alpha-5 & | & \beta-13 \end{bmatrix}$$</p><p><strong>Step 4:</strong> For infinitely many solutions, the third row must be all zeros:</p><p>$$\alpha - 5 = 0 \implies \alpha = 5$$</p><p>$$\beta - 13 = 0 \implies \beta = 13$$</p><p><strong>Step 5:</strong> Calculate β - α:</p><p>$$\beta - \alpha = 13 - 5 = 8$$</p><p>∴ Answer: <strong>8</strong></p>
Correct Answer: 8