WAEC 2016 · Paper 2 · Q8

(a)
Given that $\mathbf p = \begin{pmatrix} 5 \\ 3 \end{pmatrix}$ , $\mathbf q = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$ , $\mathbf r = \begin{pmatrix} 17 \\ 5 \end{pmatrix}$ and $\mathbf r = \alpha\mathbf p + \beta\mathbf q$ , where $\alpha$ and $\beta$ are scalars, express $\mathbf q$ in terms of $\mathbf r$ and $\mathbf p$ .
Show the answer
$\mathbf q = \frac32\mathbf p - \frac12\mathbf r$

Worked solution (try it first)

Match the parts of $\mathbf r = \alpha\mathbf p + \beta\mathbf q$ : $5\alpha - \beta = 17$ and $3\alpha + 2\beta = 5$ .
From the first, $\beta = 5\alpha - 17$ .
Substitute: $3\alpha + 10\alpha - 34 = 5$ , so $\alpha = 3$ .
Then $\beta = 15 - 17 = -2$ , so $\mathbf r = 3\mathbf p - 2\mathbf q$ .
Rearrange: $2\mathbf q = 3\mathbf p - \mathbf r$ , so $\mathbf q = \frac32\mathbf p - \frac12\mathbf r$ .