For which value of K will the equation pair x + 2y - 3 = 0 and 5x + ky +7 = 0 have no solution:

Explanation -

For a system of linear equations to have no solution, the lines they represent must be parallel and not intersect. In terms of the equations provided:

1) x + 2y - 3 = 0
2) 5x + ky + 7 = 0

These equations are in the form Ax + By + C = 0, where A, B, and C are coefficients.

The first equation can be rearranged as x + 2y = 3, and the second equation as 5x + ky = -7.

To find the value of k for which these lines are parallel and have no solution, they must have the same slope but different y-intercepts.

The slope-intercept form of the first equation x + 2y = 3 is \(y = -\frac{1}{2}x + \frac{3}{2}\).

The slope-intercept form of the second equation 5x + ky = -7 is \(y = -\frac{5}{k}x - \frac{7}{k}.\)

For the lines to be parallel, the slopes of the lines must be equal, 

So \(\frac{-1}{2}=\frac{-5}{k} \implies k =10\)

Hence the equations x + 2y - 3 = 0 and 5x + ky +7 = 0 have no solution for k = 10.