WAEC 2022 · Paper 2 · Q9

(a)
Copy and complete the table for the relation $y = \dfrac{6}{x - 1} + 1$ for $2 \le x \le 7$ .

$x$ 2 2.5 3 4 5 6 7

$y$ 7 3 2

Model answer

$x$ 2 2.5 3 4 5 6 7

$y$ 7 5 4 3 2.5 2.2 2

For example, at $x = 6$ : $y = \frac{6}{5} + 1 = 2.2$ .
(b)
Using a scale of 2 cm to 1 unit on both axes, draw the graphs of the relations: (i) $y = \dfrac{6}{x - 1} + 1$ ; (ii) $y = 8 - x$ .
Model answer
Plot every point from the table, then join them with one smooth curve (not straight lines between points). This curve is not a parabola: it drops steeply near $x = 2$ and levels off towards $y = 1$ . Draw $y = 8 - x$ through $(0, 8)$ and $(8, 0)$ .

For (c): $(x - 1)(y - 1) = 6$ is the same as $y = \frac{6}{x - 1} + 1$ , and $x + y = 8$ is $y = 8 - x$ , so the solutions are where the graphs cross: $x \approx 2.3, y \approx 5.7$ and $x \approx 5.7, y \approx 2.3$ .
(c)
Using the graphs, find, correct to two significant figures, the solutions of the simultaneous equations $(x - 1)(y - 1) = 6$ and $x + y = 8$ . (Give the two $x$ -values.)
Separate values with commas, e.g. 3, −2

$x$	2	2.5	3	4	5	6	7
$y$	7			3			2

$x$	2	2.5	3	4	5	6	7
$y$	7	5	4	3	2.5	2.2	2

The curve and the line cross at the two solutions.

Worked solution (try it first)

Rearrange the first equation: $(x - 1)(y - 1) = 6$ gives $y - 1 = \frac{6}{x - 1}$ , which is the curve in (b)(i).
And $x + y = 8$ is the line $y = 8 - x$ .
So the solutions are the points where the curve and the line cross.
Reading from the graph: $x \approx 2.3$ , $y \approx 5.7$ and $x \approx 5.7$ , $y \approx 2.3$ .
(By algebra, $x = 4 \pm \sqrt3$ .)