Question

Find a and b if $a(\begin{array}{c}2\\ 3\end{array})+b(\begin{array}{c}-1\\ 1\end{array})=(\begin{array}{c}10\\ 5\end{array})$.

Answer 1

$a\left[\begin{array}{c}2\\ 3\end{array}\right]+b\left[\begin{array}{c}-1\\ 1\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}2a\\ 3a\end{array}\right]+\left[\begin{array}{c}-b\\ b\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}2a-b\\ 3a+b\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right]$

Comparing both sides, we get

$\Rightarrow 2a-b=10⟶\left(1\right)$

$\Rightarrow 3a+b=5⟶\left(2\right)$

Adding eqations $\left(1\right)\xi \left(2\right),$ we get

$\Rightarrow 5a=15$

$\Rightarrow a=3$

Put $a=3$ in equation $\left(1\right),$ we get

$\Rightarrow 2\times 3-b=10$

$\Rightarrow 6-b=10$

$\Rightarrow b=-4$

$\Rightarrow a=3$

Hence, the answer is $a=3,b=-4.$