If p = {pmatrix}2 \\ 4{pmatrix} and q = {pmatrix}10 \\ -1{pmatrix}, find a vector, r such that 2p - 3r = q

Correct Answer is : (−23){pmatrix}-2 \\ 3{pmatrix}(−23)

Correct Answer is : (−23){pmatrix}-2 \\ 3{pmatrix}(−23)

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WAEC Further Mathematics 2024 Paper

Question : 12

Total: 40

If p =

\begin{pmatrix}2 \\ 4\end{pmatrix}

\begin{pmatrix}10 \\ -1\end{pmatrix}

, find a vector, r such that 2p - 3r = q

Solution:

Concept:

Solve the vector equation

2p - 3r = q

by writing

r

\begin{pmatrix}x \\ y\end{pmatrix}

and then comparing the corresponding components.

Explanation:

Let

r = \begin{pmatrix}x \\ y\end{pmatrix}

Given

p = \begin{pmatrix}2 \\ 4\end{pmatrix}

and

q = \begin{pmatrix}10 \\ -1\end{pmatrix}

Substitute these into the equation:

2\begin{pmatrix}2 \\ 4\end{pmatrix} - 3\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}10 \\ -1\end{pmatrix}

This simplifies to:

\begin{pmatrix}4 \\ 8\end{pmatrix} - \begin{pmatrix}3x \\ 3y\end{pmatrix} = \begin{pmatrix}10 \\ -1\end{pmatrix}

Compare the first components:

4 - 3x = 10

-3x = 6

x = -2

Compare the second components:

8 - 3y = -1

-3y = -9

y = 3

Therefore, the vector that satisfies the equation is

r = \begin{pmatrix}-2 \\ 3\end{pmatrix}

Answer:

r = \begin{pmatrix}-2 \\ 3\end{pmatrix}

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