Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Given the system of linear equations<br>\(x - 4y + 7z = g\)<br>\(3y - 5z = h\)<br>\(-2x + 5y - 9z = k\)<br>is consistent, then which of the following is correct?</p>
<p>\(g + h + k = 0\)</p>
<p>\(2g + h + k = 0\)</p>
<p>\(g + 2h + k = 0\)</p>
<p>\(g + h + 2k = 0\)</p>
Step-by-Step Solution
Key Concept: A system of linear equations is consistent if and only if the augmented matrix and coefficient matrix have the same rank. Use row operations on the augmented matrix to find the constraint relating g, h, and k.
<p><strong>Step 1:</strong> Write the augmented matrix for the system:</p><p>$$\begin{bmatrix} 1 & -4 & 7 & | & g \\ 0 & 3 & -5 & | & h \\ -2 & 5 & -9 & | & k \end{bmatrix}$$</p><p><strong>Step 2:</strong> Apply row operation R₃ → R₃ + 2R₁:</p><p>$$\begin{bmatrix} 1 & -4 & 7 & | & g \\ 0 & 3 & -5 & | & h \\ 0 & -3 & 5 & | & k+2g \end{bmatrix}$$</p><p><strong>Step 3:</strong> Apply row operation R₃ → R₃ + R₂:</p><p>$$\begin{bmatrix} 1 & -4 & 7 & | & g \\ 0 & 3 & -5 & | & h \\ 0 & 0 & 0 & | & k+2g+h \end{bmatrix}$$</p><p><strong>Step 4:</strong> For the system to be consistent, the last row must not represent a contradiction. Therefore:</p><p>$$k + 2g + h = 0$$ or equivalently $$\boxed{k + h + 2g = 0}$$</p><p>∴ Answer: B (This is the consistency condition relating g, h, and k)</p>
Correct Answer: B