The planes $2x + 3y - z = 5$ and $x - y + 2z = k$ intersect in the line $5x + 1 = 9 - 5y = - 5z$ .

Find the value of k .

point on line is $x = \frac{{ - 1 - 5\lambda }}{5}{\text{, }}y = \frac{{9 + 5\lambda }}{5}{\text{, }}z = \lambda$ or similar     M1A1

 Note: Accept use of point on the line or elimination of one of the variables using the equations of the planes

$\frac{{ - 1 - 5\lambda }}{5} - \frac{{9 + 5\lambda }}{5} + 2\lambda  = k$     M1A1

 Note: Award M1A1 if coordinates of point and equation of a plane is used to obtain linear equation in k or equations of the line are used in combination with equation obtained by elimination to get linear equation in k.

 $k = - 2$     A1

[5 marks]

Many different attempts were seen, sometimes with success. Unfortunately many candidates wasted time with aimless substitutions showing little understanding of the problem.