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Find vectors a and b such that a + b = [-2, 6] and 2a - b = [5, 0].

September 28, 2026Mathematics

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Answer

a = \begin{pmatrix} 1 \\ 2 \end{pmatrix}

b = \begin{pmatrix} -3 \\ 4 \end{pmatrix}

3 steps

Here's the solution for question 10:

We are given a system of two vector equations:

Add equation (1) and equation (2) to eliminate vector $b$ .
$(a + b) + (2a - b) = \begin{pmatrix} -2 \\ 6 \end{pmatrix} + \begin{pmatrix} 5 \\ 0 \end{pmatrix}$ $3a = \begin{pmatrix} -2+5 \\ 6+0 \end{pmatrix}$ $3a = \begin{pmatrix} 3 \\ 6 \end{pmatrix}$
Solve for vector $a$ .
$a = \frac{1}{3} \begin{pmatrix} 3 \\ 6 \end{pmatrix}$ $a = \begin{pmatrix} \frac{3}{3} \\ \frac{6}{3} \end{pmatrix}$ $a = \boxed{\begin{pmatrix} 1 \\ 2 \end{pmatrix}}$
Substitute the value of $a$ into equation (1) to find vector $b$ .
$\begin{pmatrix} 1 \\ 2 \end{pmatrix} + b = \begin{pmatrix} -2 \\ 6 \end{pmatrix}$ $b = \begin{pmatrix} -2 \\ 6 \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ $b = \begin{pmatrix} -2-1 \\ 6-2 \end{pmatrix}$ $b = \boxed{\begin{pmatrix} -3 \\ 4 \end{pmatrix}}$

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