Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>The values of <math>\lambda</math> and <math>m</math> for which the system of linear equations <math>x + y + z = 2</math>, <math>x + 2y + 3z = 5</math>, <math>x + 3y + \lambda z = m</math> has infinitely many solutions are, respectively</p>
<p>(a) 6 and 8</p>
<p>(b) 5 and 7</p>
<p>(c) 5 and 8</p>
<p>(d) 4 and 9</p>
Step-by-Step Solution
Key Concept: A system of linear equations has infinitely many solutions when the coefficient matrix and augmented matrix have the same rank, which is less than the number of unknowns. This occurs when the rows become linearly dependent, making the third equation a linear combination of the first two.
<p><strong>Step 1: Set up the system</strong></p><p>The system is:<br>x + y + z = 2 ... (1)<br>x + 2y + 3z = 5 ... (2)<br>x + 3y + λz = m ... (3)</p><p><strong>Step 2: Condition for infinitely many solutions</strong></p><p>For infinitely many solutions, the rank of coefficient matrix equals rank of augmented matrix, but both are less than 3. This means row 3 must be a linear combination of rows 1 and 2.</p><p><strong>Step 3: Find the linear combination</strong></p><p>Subtract equation (1) from (2):<br>y + 2z = 3 ... (4)</p><p>Subtract equation (1) from (3):<br>2y + (λ-1)z = m - 2 ... (5)</p><p>For the system to have infinitely many solutions, equation (5) must be a scalar multiple of equation (4).</p><p><strong>Step 4: Determine the scalar multiple</strong></p><p>If equation (5) = 2 × equation (4):<br>2y + 4z = 6</p><p>Comparing with 2y + (λ-1)z = m - 2:<br>λ - 1 = 4 ⟹ λ = 5<br>m - 2 = 6 ⟹ m = 8</p><p><strong>Step 5: Verification</strong></p><p>With λ = 5 and m = 8:<br>Row 3: x + 3y + 5z = 8<br>Row 1 + Row (4)×2: x + y + z + 2(y + 2z) = 2 + 2(3) = 8<br>This gives: x + 3y + 5z = 8 ✓</p><p><strong>∴ Answer: c</strong></p>
Correct Answer: c