Final answer:
For the system of linear equations kx + 2y = 1 and x + y = 1 to have a unique solution, the value of k must be any number except 2. This is because the two lines must have different slopes to intersect exactly once.
Step-by-step explanation:
To find the value(s) of k for which the system of linear equations kx + 2y = 1 and x + y = 1 has a unique solution, we need to consider the coefficients. For a unique solution to exist in a system of two linear equations, the lines represented by these equations must intersect exactly once. This means that their slopes must be different. The slope-intercept form of a line is y = mx + b, where m is the slope.
The second equation is already in slope-intercept form, with a slope of -1. To find the slope of the first equation, we can rearrange it to solve for y: y = (1/2) - (k/2)x. The slope here is -k/2. These two slopes must be different for the equations to have a unique solution, so we set -k/2 not equal to -1 and solve for k:
-k/2 ≠ -1
Multiplying both sides by 2:
k ≠ 2
The system of equations will have a unique solution for all values of k except when k is 2. Therefore, the correct answer is All k ≠ 2 (All k not equal to 2).