Straight Lines
Conditions for parallel/coincident lines
Grade 11
Question:
<p>For no solution, the lines represented by the system must be parallel. Given the system of equations with parameter <em>k</em>, the condition for no solution gives \(k^2 - 4k + 3 = 0\), so \(k = 1, 3\). Since \(k = 1\) gives coincident lines, how many values of <em>k</em> give no solution?</p>
<p>0</p>
<p>1</p>
<p>2</p>
<p>3</p>
Step-by-Step Solution
Key Concept: For a system of linear equations to have no solution, the lines must be parallel (same slope, different intercepts). Coincident lines (k=1) represent infinitely many solutions, not no solution, so must be excluded from the count.
<p><strong>Step 1:</strong> For no solution in a system of equations, we need parallel but distinct lines (same slope, different y-intercepts).</p><p><strong>Step 2:</strong> The condition k² - 4k + 3 = 0 factors as (k-1)(k-3) = 0, giving k = 1 or k = 3.</p><p><strong>Step 3:</strong> Check k = 1: This makes the lines coincident (identical equations), yielding infinitely many solutions, NOT no solution.</p><p><strong>Step 4:</strong> Check k = 3: This makes the lines parallel and distinct, yielding no solution. ✓</p><p><strong>Step 5:</strong> Exclude k = 1 since it gives coincident lines (infinitely many solutions).</p><p>∴ Answer: <strong>1 value of k</strong> (only k = 3) gives no solution. <strong>B</strong></p>
Correct Answer: B