Simultaneous equations and planesEdexcel A-Level Further Maths: Flashcards

What these 12 flashcards ask

How do you write three simultaneous equations as a matrix equation?
How do you solve \mathbf{M}\mathbf{x}=\mathbf{b} using the inverse?
What does \det\mathbf{M}\ne0 tell you about the planes?
What does \det\mathbf{M}=0 tell you about the equations?
What is a sheaf of planes?
What is a prism configuration?
How can you tell that two planes are parallel?
How can you tell that two planes are coincident?
What is the normal vector to ax+by+cz=d?
Planes with proportional normals but constants not in that ratio?
If the third equation is the sum of the first two (constants too), what are the planes?
If the left-hand side of the third is the sum of the first two but the constant is not?

Exam questions on Simultaneous equations and planes

Three planes have equations $x+y+z=6$ , $\ 2x-y+z=3$ and $x+2y-z=2$ .
Use the inverse of $\mathbf{M}$ , found using your calculator, to solve the equations.2 marks
Three planes have equations $x+2y-z=4$ , $\ 2x+y+z=5$ and $3x+3y=k$ , where $k$ is a constant.
Describe the geometrical configuration of the three planes when $k=6$ , giving a reason.2 marks
Three planes have equations $\Pi_1:\ x+2y+3z=5$ , $\ \Pi_2:\ 2x+4y+6z=7$ and $\Pi_3:\ x-y+z=1$ .
Explain why the equations have no solution, and describe the geometrical configuration of the three planes.3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).