a and b are vectors such that \[ a = \begin{pmatrix} 2 \\ -3 \end{pmatrix} \] and \[ 3a - 2b = \begin{pmatrix} -8 \\ -17 \end{pmatrix} \] Find b as a column vector. - Edexcel - GCSE Maths - Question 14 - 2022 - Paper 3

First, calculate the vector (3a):

[ 3a = 3 \cdot \begin{pmatrix} 2 \ -3 \end{pmatrix} = \begin{pmatrix} 6 \ -9 \end{pmatrix} ]

Now substitute (3a) into the equation:

[ \begin{pmatrix} 6 \ -9 \end{pmatrix} - 2b = \begin{pmatrix} -8 \ -17 \end{pmatrix} ]

Rearranging for (2b) gives:

[ 2b = \begin{pmatrix} 6 \ -9 \end{pmatrix} + \begin{pmatrix} -8 \ -17 \end{pmatrix} ]

Now perform the addition:

[ 2b = \begin{pmatrix} 6 - 8 \ -9 - 17 \end{pmatrix} = \begin{pmatrix} -2 \ -26 \end{pmatrix} ]

Next, divide both sides by 2 to find (b):

[ b = \begin{pmatrix} -2 \div 2 \ -26 \div 2 \end{pmatrix} = \begin{pmatrix} -1 \ -13 \end{pmatrix} ]