Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>If \(u + 2v + 3w = 6\), \(4u + 5v + 6w = 12\), and \(6u + 9v = 4\), then \(u + v + w\) is equal to</p>
<p>(a) 2</p>
<p>(b) \(\frac{1}{2}\)</p>
<p>(c) 20</p>
<p>(d) \(\frac{1}{20}\)</p>
Step-by-Step Solution
Key Concept: Solve the system of three linear equations by elimination to find u, v, and w, then sum them.
<p><strong>Solution:</strong></p><p>Given equations:</p><p>$u + 2v + 3w = 6$ ...(i)</p><p>$4u + 5v + 6w = 12$ ...(ii)</p><p>$6u + 9v = 4$ ...(iii)</p><p>From (i) and (ii):</p><p>Subtracting 4×(i) from (ii): $-3v - 6w = -12$, which gives $2u + v = 0$ ...(iv)</p><p>Solving (iii) and (iv):</p><p>From (iv): $v = -2u$</p><p>Substituting in (iii): $6u + 9(-2u) = 4$</p><p>$6u - 18u = 4$</p><p>$u = -\frac{1}{3}$, $v = \frac{2}{3}$</p><p>From (i): $w = \frac{5}{3}$</p><p>$u + v + w = -\frac{1}{3} + \frac{2}{3} + \frac{5}{3} = 2$</p>
Correct Answer: a