Question

Carlos writes three linear equations in different forms. He says they are equivalent equations. How could you verify that he is correct?

a) Check if the equations have the same coefficients and constants.
b) Verify if the equations have the same solutions when graphed.
c) Determine if the equations have the same number of variables.
d) Confirm if the equations represent the same physical system.

Answer 1

To verify if three linear equations are equivalent, you should graph them and see if they overlap, indicating they have the same set of solutions.

Step-by-step explanation:

To verify if three linear equations written by Carlos in different forms are equivalent, one should verify if the equations have the same solutions when graphed (option b). If the equations are equivalent, they will all intersect at the same points, and the graph of each equation will overlap exactly on top of one another. Checking if the equations have the same coefficients and constants (option a) might not always be sufficient, as equivalent equations can still be expressed with different coefficients due to simplification or multiplication by a common factor. The number of variables (option c) should be the same in all three equations if they are linear, but that alone does not confirm equivalence. Finally, confirming if the equations represent the same physical system (option d) is unnecessary in this mathematical context. Instead, you would graph the equations or solve them algebraically to see if they yield the same set of solutions.