2直線の交点を通る直線の方程式

\(\begin{eqnarray}~~~k\left\{2\cdot(-2)+(-2)-1\right\}+\left\{(-2)+4\cdot(-2)+3\right\}&=&0
\\[3pt]~~~k(-4-2-1)+(-2-8+3)&=&0
\\[3pt]~~~-7k-7&=&0
\\[3pt]~~~-7k&=&7
\\[3pt]~~~k&=&-1\end{eqnarray}\)

\(\begin{eqnarray}~~~-1\cdot(2x+y-1)+(x+4y+3)&=&0
\\[3pt]~~~-2x-y+1+x+4y+3&=&0
\\[3pt]~~~-x+3y+4&=&0
\\[3pt]~~~x-3y-4&=&0\end{eqnarray}\)

\(\begin{eqnarray}~~~k\left\{2\cdot(-2)-(1)+1\right\}+\left\{(-2)+(1)-4\right\}&=&0
\\[3pt]~~~k(-4-1+1)+(-2+1-4)&=&0
\\[3pt]~~~-4k-5&=&0
\\[3pt]~~~-4k&=&5
\\[3pt]~~~k&=&-\displaystyle\frac{\,5\,}{\,4\,}\end{eqnarray}\)

\(\begin{eqnarray}~~~-\displaystyle\frac{\,5\,}{\,4\,}(2x-y+1)+(x+y-4)&=&0
\\[5pt]~~~-5(2x-y+1)+4(x+y-4)&=&0~~~\hspace{10pt}(\,∵~{\, \small \times \,}4\,)
\\[3pt]~~~-10x+5y-5+4x+4y-16&=&0
\\[3pt]~~~-6x+9y-21&=&0
\\[3pt]~~~2x-3y+7&=&0~~~\hspace{10pt}(\,∵~{\, \small \div \,}(-3)\,)\end{eqnarray}\)

\(\begin{eqnarray}~~~k(4\cdot1-5\cdot1+5)+(1+2\cdot1-6)&=&0
\\[3pt]~~~k(4-5+5)+(1+2-6)&=&0
\\[3pt]~~~4k-3&=&0
\\[3pt]~~~4k&=&3
\\[3pt]~~~k&=&\displaystyle\frac{\,3\,}{\,4\,}\end{eqnarray}\)

\(\begin{eqnarray}~~~\displaystyle\frac{\,3\,}{\,4\,}(4x-5y+5)+(x+2y-6)&=&0
\\[5pt]~~~3(4x-5y+5)+4(x+2y-6)&=&0~~~\hspace{10pt}(\,∵~{\, \small \times \,}4\,)
\\[3pt]~~~12x-15y+15+4x+8y-24&=&0
\\[3pt]~~~16x-7y-9&=&0\end{eqnarray}\)