How would you know if a linear system has a solution?

Answer 1

One way is to look at the graphs of these equations. If they intersect, the point of intersection (x, y) is the only solution of the system. In this case we say that the system is consistent.

If their graphs do not intersect, then the system has no solution. In this case we say that the system is inconsistent.

If the graph of the equations is the same line, the system has infinitely simultaneous solutions.

We can use several methods in order to solve the system algebraically.

In the case where the equations of the system are dependent (the coefficients of the same variable are multiple of each other), the system has infinite number of solutions solution. For example,

2x + 3y = 6

4y + 6y = 12

These equations are dependent. Since they represent the same line, all points that satisfy either of the equations are solutions of the system.

Try to solve this system of equations,

2x + 3y = 6

4x + 6y = 7

If you use addition or subtraction method, and you obtain a peculiar result such that 0 = 5, actually you have shown that the system has no solution (there is no point that satisfying both equations).

When you use the substitution method and you obtain a result such that 5 = 5, this result indicates no solution for the system.