Question

For what value of \(k\) are the two lines \(2 x+k y=3\) and \(x+y=1\) (a) parallel? (b) perpendicular?

Short answer

For the lines to be parallel, \(k = 2\). For the lines to be perpendicular, \(k = 2\). The result is essentially the same in this special case.

Step-by-step solution

Convert to slope-intercept form

Let's convert both equations into slope-intercept form. \n For \(2x + ky = 3\), we get \(y = (-2/k)x + (3/k)\). So, the slope is \(-2/k\). \n For \(x+y=1\), we get \(y = -x + 1\). So, the slope is -1.

Find \(k\) for the lines to be parallel

For the lines to be parallel, their slopes must be equal. Thus, we equate \(-2/k\) and -1 which gives us \(k = -2/k = -1\). Therefore, \(k = 2\). This is the value of \(k\) when the two lines are parallel.

Find \(k\) for the lines to be perpendicular

For the lines to be perpendicular, the product of their slopes must be -1. Thus, we find \(k\) such that \(-2/k * -1 = -1\). Solving, we find \(k = 2\). Therefore, \(k = 2\) is the value of k when the two lines are perpendicular.