Find the equation of the plane through $(-1,4,2)$ and containing the line of intersection of the planes $$4x-y+z-2=0 \\ 2x+y-2z-3=0$$
My answer comes out to be: $$-9x-67y+104=51$$ While the answer provided on the answer sheet is: $$4x-13y+21z=-14$$ Could you please check if the answer calculated by me is correct or the one provided on the answer sheet.
Other hint
The equation of sheaf of planes passing through the intersection of planes 4x-y+z-2=0 and 2x+y-2z-3=0:
$\lambda_1(4x-y+z-2)+\lambda_2(2x+y-2z-3)=0$. The specific plane of sheaf of planes determine numbers $\lambda_1,\,\lambda_2$, which are not simultaneously = $0$.
The equation of the plane through (−1,4,2) --> coordinates put into equation of sheaf of planes:
$\displaystyle \lambda_1(-4-4+ 2-2)+\lambda_2(-2+4-4-3)=0\Rightarrow \frac{\lambda_1}{\lambda_2}=-\frac{5}{8}$
$\displaystyle -\frac{5}{8}(4x-y+z-2)+(2x+y-2z-3)=0\Rightarrow \cdots \Rightarrow 4x-13y+21z+14=0$
Notice, the equation of the plane passing through the line of intersection of planes: $4x-y+z-2=0$ & $2x+y-2z-3=0$ is given as $$4x-y+z-2+\lambda(2x+y-2z-3)=0$$ $$(2\lambda+4)x+(\lambda-1)y+(1-2\lambda)z=(3\lambda+2)\tag 1$$ Since, the above plane passes through the given point $(-1, 4, 2)$, hence it will satisfy the equation (1) of the plane as follows $$(2\lambda+4)(-1)+(\lambda-1)(4)+(1-2\lambda)(2)=(3\lambda+2)$$ $$-5\lambda-8=0\implies \lambda=\frac{-8}{5}$$ Now, substituting the value of $\lambda$ in (1), we get the equation of the plane as follows $$\left(2\left(\frac{-8}{5}\right)+4\right)x+\left(\frac{-8}{5}+4\right)y+\left(1-2\left(\frac{-8}{5}\right)\right)z=\left(3\left(\frac{-8}{5}\right)+2\right)$$ $$\color{blue}{4x-13y+21z=-14}$$
HINT:
$$\begin{cases} 4x-y+z-2=0\\ 2x+y-2z-3=0 \end{cases}\Longleftrightarrow$$
$$\begin{cases} z=-4x+y+2\\ 2x+y-2z-3=0 \end{cases}\Longleftrightarrow$$
$$\begin{cases} z=-4x+y+2\\ 2x+y-2(-4x+y+2)-3=0 \end{cases}\Longleftrightarrow$$
$$\begin{cases} z=-4x+y+2\\ 10x-y-7=0 \end{cases}\Longleftrightarrow$$
$$\begin{cases} z=-4x+y+2\\ y=10x-7 \end{cases}\Longleftrightarrow$$
$$\begin{cases} z=-4x+(10x-7)+2\\ y=10x-7 \end{cases}\Longleftrightarrow$$
$$\begin{cases} z=6x-5\\ y=10x-7 \end{cases}$$
