A system of two linear equations has no solution when no ordered pair satisfies both equations, represented graphically by distinct parallel lines with the same slope but different intercepts. Algebraically, simplifying or eliminating a variable produces a false statement such as , distinguishing this case from coincident lines, which represent infinitely many solutions; this interpretation supports reasoning about consistency and linear models.
To identify a system with no solution, simplify the equations and try to eliminate one variable. If you get a false statement, such as , the system has no solution.
Consider:
Step 1: Compare the equations.
The left side of the second equation is twice the left side of the first:
If the equations represented the same line, the right side would also be twice:
But the second equation says the right side is , not .
Step 2: Eliminate a variable.
Multiply the first equation by :
Add it to the second equation:
The statement is false, so there is no possible value of and that makes both equations true.
Therefore, the system has no solution.
Graphically, the equations represent distinct parallel lines: they have the same slope but different intercepts, so they never intersect.
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