<p>If the following system of equations is consistent,<br>\((a+1)^3x + (a+2)^3y = (a+3)^3\)<br>\((a+1)x + (a+2)y = a+3\)<br>\(x + y = 1\),<br>then find the value of \(a\).</p>
Step-by-Step Solution
Key Concept: For a system of 3 equations in 2 unknowns to be consistent, the equations must be dependent (proportional). Check if the third equation is a linear combination of the first two by comparing ratios of coefficients.
<p><strong>Step 1:</strong> For the system to be consistent with 3 equations in 2 unknowns, the equations must be linearly dependent. The third equation x + y = 1 must be a linear combination of the first two.</p><p><strong>Step 2:</strong> Assume equation (3) = λ × equation (2) + μ × equation (1):</p><p>x + y = 1 = λ[(a+1)x + (a+2)y] + μ[(a+1)³x + (a+2)³y]</p><p><strong>Step 3:</strong> Comparing coefficients of x: 1 = λ(a+1) + μ(a+1)³</p><p>Comparing coefficients of y: 1 = λ(a+2) + μ(a+2)³</p><p>Comparing constants: 1 = λ(a+3) + μ(a+3)³</p><p><strong>Step 4:</strong> For consistency, test if equation (2) is proportional to equation (3), or if a specific value of a makes the first equation proportional to the second and third.</p><p><strong>Step 5:</strong> If a+1, a+2, a+3 are consecutive integers, check if the system becomes dependent when a = -2:</p><p>• When a = -2: equations become -1·x + 0·y = 1, -x + 0·y = 1, x + y = 1</p><p>• The first two equations give x = -1, which with x + y = 1 gives y = 2 ✓</p><p>• Verify: (-1)³(-1) + 0(0) = 1 ✓, -(-1) + 0 = 1 ✓, -1 + 2 = 1 ✓</p><p>∴ Answer: <strong>a = -2</strong></p>
Correct Answer: -2