is a linear equation.
is NOT a linear equation.
is a linear equation. (should be written as .)
(This is known as the standard form)
In general, we will be looking at systems of linear equations in variables:
Where is the coefficient of variable in the equation .
is a solution of the system if the equations are satisfied when
A system of linear equations is consistent if there is at least one solution.
Otherwise it is inconsistent.
The system is consistent.
Geometrically, we have two planes in that intersect. The point is one of the points of intersection.
No solutions. System is inconsistent.
Geometrically, we have two lines in that are parallel and thus do NOT intersect.
In general, a solution of a system of linear equations in variables is a point in where hyperplanes represented
by the linear equations, intersect.
We could have a system with
No solution. (Inconsistent)
Eg parallel hyper planes.
All the hyperplanes intersect at a single point.
Eg Intersection of the planes is a line.
Can you create a system with exactly 2 solutions?