Abc is an isosceles right triangle in which ab has a slope of -1 and angle abc = 90 degrees. triangle abc is dilated by a scale factor of 1.8 with the origin as the center of dilation, resulting in the image a'b'c'. what is the slope of b'c'?

The dilation by a scale factor of 1.8 means that every x and y coordinate are multiplied by 1.8 converting x,y into 1.8x, 1.8y.

This kind of transformation results in a similar figure.

So the triangle A'B'C' will be similar to triangle ABC: which means that all the angles are preserved (thy are congruent).

Being the angles congruent the slope of B'C' will be the same angle of BC.

Since, the slope of AB is - 1, the other two sopes are one 0 and the other undefined (∞). The horizontal side has slope 0 and the vertical side has slope ∞.
Since you did not attache any figure I do not know whether B'C' is the horizontal or the vertical side.

The answer is that the slope of B'C' is zero if it is the horizontal side, and the slope is ∞ if B'C' is the vertical side.

Yes, these pairs are linear because there is only one x value per y value. Or in other words, no repeats!