Find scalars so that a given vector equation is satisfied

by RoRi

April 28, 2016

Let

$A = (2,-1,1), \qquad B = (1,2,-1), \qquad C = (2,-11,7).$

Find $x,y \in \mathbb{R}$ such that $C = xA + yB$ .

Since $C = xA + yB$ we obtain the three equations

$\begin{align*} 2x+y &= 2 \\ -x + 2y &= -11 \\ x - y &= 7. \end{align*}$

From the third equation we have $x = y+ 7$ . Plugging this into the second equation we have $-y-7+2y = -11$ which gives us $y = -4$ . This implies $x = 3$ . Checking that these values of $x,y$ still satisfy the first equation we have $2(3) + (-4) = 6 - 4 = 2$ . So, indeed, $x = 3$ , $y = -$ is a solution to this equation.