How can you tell that a system of linear equations will have infinitely many solutions without computation?

A system of linear equations will have infinitely many solutions if the coefficients can be manipulated so that they are the same for all variables.

According to the question;

For instance; if we have equations;

The second equation can be manipulated so that it is similar to the first by dividing through by 3.

As such, the system of equations will have infinitely many solutions.

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If you can manipulate the two equations so that they have exactly the same coefficients (e. g., 3x + 4y = 8 and 2x + 4y = 8), then you conclude that the two lines coincide (overlap), and that there are thus infinitely many solutions.

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