Answer
(a)$$y=mx$$
(b)$$y=mx-m$$
(c)$$y=mx-m-2$$
(d)$$y=-\frac{1}{2}x+b$$
Work Step by Step
The equation of a line with slope $m$ and $y$-intercept $b$ is$$y=mx+b.$$
(a) The lines passing through the origin have $b=0$, so an equation of the family lines passing through the origin is $$y=mx.$$
(b) For the lines with $x$-intercept $a=1$ we have$$0=m(1)+b \quad \Rightarrow \quad b=-m.$$Thus, an equation of the family of lines with $x$-intercept $a=1$ is $$y=mx-m.$$
(c) For the lines passing through through the point $(1,-2)$ we have$$-2=m(1)+b \quad \Rightarrow \quad b=-m-2.$$Thus, an equation for the family of lines passing through through the point $(1,-2)$ is$$y=mx-m-2.$$
(d) The lines parallel to the line$$2x+4y=1 \quad \Rightarrow \quad y= - \frac{1}{2}x+ \frac{1}{4}$$have slope $m=-\frac{1}{2}$, so an equation for the family of lines parallel to $2x+4y=1$ is$$y=-\frac{1}{2}x+b.$$
