Describing Solutions to Systems of Linear Equations

A system of linear equations can either have one solution, infinitely many solutions or no solution. Let's think about what leads us to having these different results.

• Infinitely many solutions: The equations are the exact same line (they have the same slope and $y\text{-}$intercept). There are infinitely many solutions because the lines will intersect at every single possible point.
• No solution: The equations are parallel lines (they have the same slope but different $y\text{-}$intercepts). There are no solutions because the lines will never intersect.
• One solution: The equations are nonparallel lines (they have different slopes). These lines will intersect exactly once due to the continuous nature of linear functions.

The equations John tricked Hans with have slopes of $3$ and $\text{-}2.$ These are going to be nonparallel lines and will, therefore, intersect one time and have only one solution.