question is.For what value of K are the two lines 2x + Ky = 3 and x + y = 1 (a) parallel?(b) perpendicular?

y = -x + 1

slope is -1.

parrallel lines have the same slope. So the slope of 2x + Ky = 3 must be also -1

solve for y

2x + Ky = 3

Ky = -2x + 3

y = -2x/K + 3

take the part of -2/K

we want the slope equals -1

so, -2/K = -1

K = 2

so the value of K must be 2 so that the line 2x + Ky = 3 is parrellel to x + y = 1

the slope that is perpendicular to x + y = 1 is OPPOSITE RECIPROCAL to the slope of that line.

again, x + y = 1 has the slope of -1

opposite of -1 is 1,

the reciprocal of 1 is also 1

so the slope is 1

-2/K = 1

K = -2

so the value of K must be -2 so that 2x + Ky = 3 is perpendicular to x + y = 1