How do you determine without graphing that a system of equations has no solution

Question

Trying to figure it out?

Answer 1

Two equations have parallel lines (no solution to the system) if the slopes are equal and and y-intercepts are not.

Answer 2

If it is a system of linear equations, you can evaluate the principal determinant of the system.  There is a solution if the principal determinant is not zero.

Answer 3

If two lines intersect then there is a single point at which they cross...

That point (x, y) satisfies both equations.  If there is no solution then

following the normal steps of elimination or substitution results in an

obviously false mathematical statement. Take for example two parallel lines

(same slope but different y-intercept)

 

y = 3x - 5

y = 3x + 4

 

Rearranging: 

3x - y = 5

3x - y = -4    we can multiply this by -1 and add to eliminate y

---------------      

 

 3x - y = 5

-3x + y = 4

---------------

0 + 0 = 9

0 = 9

 

Adding the equations gives an obviously false statement.

This system of equations has no solution.