How can you determine the type of solution to a system of linear equations WITHOUT graphing, using substitution, or elimination?

Answer 1

If you see that the slopes are different then there is a unique solution.

If the slopes are identical and one of the equations is a multiple of the other they are actually both the same equation which means there are infinite solutions.

Finally, if the slopes are identical but one of the equations is not a multiple of the other they are parallel lines which means there is no solution.

Answer 2

use matrix or linear algebra and row operations,

you get that in a more advanced math class, usually after calculus

or maybe you meant "type" of solution as either infinite, unique or non-existent

if the system of equations have contradictory equations, the solution DNE, i.e. Does Not Exist

graph them, they are parallel lines that never intersect

if the equations are linearly dependent, one is the same as another, there're an infinite number of equations. Graph them, they're the same line with infinite points satisfying both

otherwise, there's a unique solution

x=1

x=2

have no solution

they're contradictory

no value of x will make both equations true

graphically they're 2 parallel vertical lines that never intersect

x=1,

y=2

has a unique solution (1,2)

only one intersection point

where a vertical and horizontal line intersect

a unique solution

x+y=3

2x+2y = 6

have an infinite number of solutions

graph them, they're the same identical line