The problem may be just the syntax. I show below two ways to generate the desired plots.
(* Clear memory *)
Clear["Global`*"];
(* To generate and grid two static plots. *)
f[x_, k_, t_] := k x + t;
(* A static plot calls for a contourplot so we can vary x and t at the same time. *)
plot[k_, tlim_] := ContourPlot[ f[x, k, t],{t,-tlim,tlim},{x,-5,5}, ContourShading -> None ]
(* Call each user defined (lowercase)plot function with a specific k and limiting t(tlim) *)
Grid[{ {plot[0, 2], plot[1, 3]} }]
(* Using Manipulate. This time "t" is fixed (for convenience) to the larger range for all "k". *)
Manipulate[ Plot[ f[x,k,t], {x,-5,5},
PlotRange -> {Automatic, {-4,4}}](* Choose the range from observation. *)
{k,0,1,1,RadioButton}(* Specify k with a radio button *),
{t,-3,3,Slider}(* Specifiy t with a slider *)]
