Explanation:
The solution set for a system of equation is the set of points where the graphs of the equations intersect.
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general case
A system will have one solution if there is a single point of intersection of the graphs of the equations.
A system will have no solutions if the graphs have no points of intersection.
A system will have an infinite number of solutions if the graphs intersect at an infinite number of points.
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linear equations
When the equations are linear equations, their graphs are straight lines. If the lines have different slopes, they must intersect at exactly one point: there will be one solution.
If the lines have the same slope, there are two possibilities:
- the lines are parallel -- no solutions
- the lines are coincident -- infinite solutions
The attached graph illustrates these cases.
- the red and blue lines are the graphs of a system of equations with one solution. Those lines have different slopes
- the blue and green lines are the graphs of a system of equations with no solution. Those lines are parallel.
- The red and (dotted) purple lines are the graphs of a system of equations with infinite solutions. Those lines are coincident.
