Infinitely many solutions occur when two linear equations in two variables represent the same line: their coefficients and constants are proportional, and simplifying or eliminating a variable produces an identity such as . The learner interprets the solution set as every point on that shared line and distinguishes this case from intersecting lines with one solution and distinct parallel lines with none; higher-dimensional systems and abstract generalizations are not included.
Two linear equations have infinitely many solutions when they describe the same line. This happens when all the terms in one equation are proportional to the terms in the other.
Consider:
The second equation is the first equation multiplied by :
which gives
So both equations represent the same line.
Multiply the first equation by :
Now add it to the second equation:
The result is the identity . This means the equations do not give two different conditions. They are the same condition.
Every point on the shared line is a solution. Solving the first equation for gives:
Therefore, the system has infinitely many solutions: every point on
solves both equations. For example, and are both solutions.
When elimination produces , the system has infinitely many solutions. By contrast, intersecting lines give one solution, while distinct parallel lines give no solution.
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