Step-by-Step Solution
Key Concept: Case study on linear equations in two variables.
(a) Write the equations for upstream speed and downstream speed in terms of $x$ and $y$. [1 Mark]
Upstream speed $= x - y$ km/h; Downstream speed $= x + y$ km/h. [1.0 Mark]
(b) Form a pair of linear equations representing the two journeys using $u = \dfrac{1}{x-y}$ and $v = \dfrac{1}{x+y}$. [1 Mark]
$30u + 44v = 10$ and $40u + 55v = 13$. [1.0 Mark]
(c) Solve for $u$ and $v$. [1 Mark]
$120u + 176v = 40$ and $120u + 165v = 39 \Rightarrow 11v = 1 \Rightarrow v = 1/11$. $u = 1/5$. [1.0 Mark]
(d) Find the speed of the boat in still water ($x$) and speed of the stream ($y$). [1 Mark]
$x - y = 5, x + y = 11 \Rightarrow 2x = 16 \Rightarrow x = 8$ km/h, $y = 3$ km/h. [1.0 Mark]
Correct Answer: Upstream speed $= x - y$ km/h; Downstream speed $= x + y$ km/h. [1.0 Mark] | $30u + 44v = 10$ and $40u + 55v = 13$. [1.0 Mark] | $120u + 176v = 40$ and $120u + 165v = 39 \Rightarrow 11v = 1 \Rightarrow v = 1/11$. $u = 1/5$. [1.0 Mark] | $x - y = 5, x + y = 11 \Rightarrow 2x = 16 \Rightarrow x = 8$ km/h, $y = 3$ km/h. [1.0 Mark]