Find scalars that satisfy a given vector equation

Let

$A = (1,2), \qquad B = (2,-4), \qquad C = (2,-3).$

Find $x,y \in \mathbb{R}$ such that $C = xA + yB$ . How many pairs $(x,y)$ are there that satisfy this equation?

First, $A$ and $B$ are linearly independent since $c_1 A + c_2 B = 0$ implies

$c_1 + 2c_2 = 0 \quad \text{and} \quad 2c_1 - 4c_2 = 0$

which only has the solution $c_1 = c_2 = 0$ . Therefore, $A$ and $B$ form a basis for $\mathbb{R}^2$ so there is a unique pair $(x,y)$ such that $C = xA + yB$ . To compute them we have the two equations

$x+2y = 2 \qquad \text{and} \qquad 2x-4y = -3.$

From the first equation we have $x = 2 - 2y$ . Therefore from the second equation we have

$2(2-2y) - 4y = -3 \quad \implies \quad y = \frac{7}{8} \quad \implies \quad x = \frac{1}{4}.$

Stumbling Robot

Find scalars that satisfy a given vector equation

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