Question

Given that the vectors $\overline{a}$ and $\overline{b}$ are non-collinear, the values of $x$ and $y$ for which the equality $2\overline{u}-\overline{v}=\overline{w}$ holds where $\overline{u}=x\overline{a}+2y\overline{b},\overline{v}=-2y\overline{a}+3x\overline{b}$ and $\overline{w}=4\overline{a}-2\overline{b}$ are

• A. $x=2,y=3$
• B. $x=\frac{8}{7},y=\frac{7}{8}$
• C. $x=\frac{10}{7},y=\frac{4}{7}$
• D. $x=\frac{2}{7},y=\frac{4}{7}$

Answer 1

Answer: (C)

$x=\frac{10}{7},y=\frac{4}{7}$

We have $2\overline{u}-\overline{v}=\overline{w}$

Where

$\overline{u}=x\overline{a}+2y\overline{b},$

$\overline{v}=-2y\overline{a}+3x\overline{b}$,

$\overline{w}=4\overline{a}-2\overline{b}$

So, $\overline{a}(2x+2y)+\overline{b}(4y-3x)=4\overline{a}-2\overline{b}$

Comparing the coefficients since $\overline{a}$ and $\overline{b}$ are non collinear, we get
$2x+2y=4$ and $3x-4y=2$

Multiplying the first equation by $2$ and adding, we get

$7x=10$
$\Rightarrow x=\frac{10}{7},y=\frac{4}{7}$