Simultaneous equations and planesEdexcel A-Level Further Maths: Revision notes

Three linear equations in $x,y,z$ can be written as a matrix equation $\mathbf{M}\mathbf{x}=\mathbf{b}$, where $\mathbf{M}$ holds the coefficients, $\mathbf{x}=\begin{pmatrix}x\\ y\\ z\end{pmatrix}$ and $\mathbf{b}$ holds the right-hand sides. If $\mathbf{M}$ is non-singular ($\det\mathbf{M}\ne0$), multiply both sides on the left by $\mathbf{M}^{-1}$: $\mathbf{x}=\mathbf{M}^{-1}\mathbf{b}.$ Example: $x+y+z=6$, $2x-y+z=3$, $x+2y-z=2$ has $\det\mathbf{M}=7$ and solution $x=1$, $y=2$, $z=3$; you can check this in all three equations. Use your calculator to find $\mathbf{M}^{-1}$ in the exam, but write the matrix equation down so the method is clear.

If $\det\mathbf{M}=0$ the matrix is singular and has no inverse, so the equations do not have a unique solution. There are then two possibilities:

• the equations are consistent and have infinitely many solutions, or
• the equations are inconsistent and have no solution. To tell which, look for a dependency. For $x+2y-z=4$, $2x+y+z=5$, $3x+3y=k$ the first two equations add to $3x+3y=9$. So $k=9$ is consistent (infinitely many solutions) and any other $k$ is inconsistent (no solutions). The determinant is $0$ for every $k$.

Each linear equation $ax+by+cz=d$ is a plane in 3-D, with normal vector $(a,b,c)$. Solving three equations means finding the points common to three planes. Two planes are parallel when their normals are proportional: $(1,2,3)$ and $(2,4,6)$. They are coincident (the same plane) if the constants are in the same ratio too ($2\times5=10$), and distinct parallel planes if not ($2\times5\ne7$). If $\det\mathbf{M}\ne0$ the three planes meet at exactly one point. If $\det\mathbf{M}=0$ the planes are in one of the special configurations described next.

Consistent cases (infinitely many solutions, $\det\mathbf{M}=0$):

• Sheaf: three distinct planes sharing a common line, like pages meeting at a book's spine. The normals are not parallel. Example: $x+2y+z=1$, $2x+5y+3z=4$, $3x+7y+4z=5$ meet in the line $(-3,2,0)+t(1,-1,1)$.
• Two coincident planes and a third crossing them in a line, or all three coincident (a plane of solutions). Inconsistent cases (no solution):
• Prism: no two planes parallel, but the planes meet pairwise in three parallel lines, with no common point. Example: $x+2y-z=4$, $2x+y+z=5$, $3x+3y=6$.
• Two parallel distinct planes cut by a third, or three distinct parallel planes, or two coincident planes with a parallel third. A single point arises only when $\det\mathbf{M}\ne0$.