WAEC 2012 · Paper 2 · Q4

(a)
Indicate by shading graphically the set of all points $P(x, y)$ in the $Oxy$ plane that satisfy simultaneously the inequalities $2x - y \ge -4$ , $x + y \le 10$ , $y - x > 0$ , $y \ge 2$ and $x \ge 0$ .
Model answer
Draw each boundary line, then shade the side that satisfies every inequality. The region $R$ has corners $(5, 5)$ , $(2, 8)$ , $(0, 4)$ , $(0, 2)$ , $(2, 2)$ . The line $y = x$ is dashed because $y - x > 0$ is strict (points on it are not included). WAEC accepts either shading the wanted region or shading the unwanted side, as long as you label the region clearly.

For (b): check $10x + 5y$ at each corner; the largest value, 75, is at $(5, 5)$ .
(b)
Using the graph, find the values of $x$ and $y$ for which $10x + 5y$ is maximum.
Separate values with commas, e.g. 3, −2

The feasible region; test the corners in 10x + 5y.

Worked solution (try it first)

Draw $2x - y = -4$ , $x + y = 10$ , $y = x$ (dashed, since $y - x > 0$ ), $y = 2$ and $x = 0$ , and shade the region that satisfies all five.
Its corners: $(0, 2)$ .
$y = 2$ and $y = x$ give $(2, 2)$ .
$y = x$ and $x + y = 10$ give $(5, 5)$ .
$x + y = 10$ and $2x - y = -4$ give $3x = 6$ , so $(2, 8)$ .
$2x - y = -4$ and $x = 0$ give $(0, 4)$ .