Recall that a solution to a system of a linear equation in $n$ variables is an ordered $n$-tuple denoted $(x_1, x_2, ..., x_n) = (s_1, s_2, ..., s_n)$, which is a point of intersections between all of the equations in the system.
We stated earlier that a system of linear equations can have either one solution, infinitely many solutions, or no solutions. We will prove this later on, but until then, we will look at the various cases we can run into when dealing with solutions to systems of 2 variables and of 3 variables.
Solutions to Systems of 2 Variables
Consider a system of 2 linear equations in two variables $x, y$.
No Solutions
| 1. Two parallel lines with no intersections between lines. |
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One Solution
| 2. Two lines with a single point of intersection. |
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Infinite Solutions
| 3. Two lines that are coincident (the same) with every point on the lines being an intersection. |
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Solutions to Systems of 3 Variables
Consider a system of 3 linear equations in three variables $x, y, z$. Similarly to systems of 2 variables, systems of 3 variables can have one of three different outcomes when it comes to the number of solutions the system has, that is: no solutions, one solution, or infinitely many solutions as illustrated:
No Solutions
| 1. Three parallel planes with no intersections between planes. |
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| 2. Two parallel planes and one plane that intersects them. |
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| 3. No common intersection between all three planes. |
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| **4. ** Two equations represent the same plane while the third plane is parallel to them. |
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One Solution
| 1. All three planes intersect at a common point. |
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Infinite Solutions
| 1. All three planes intersect at a common line. All points on that line are solutions to the system. |
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| 2. Two planes are coincident while the third plane intersects them at a line. All points on that line are solutions to the system. |
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| 3. All three planes are coincident and every point on the plane is a solution to the system. |
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