telling how many solutions a system has

Question

tell whether the system has one solution, infinitely many solutions, or no solution  x+2y=3, 4x+y=19

Answer 1

Jon P. is correct. If the slopes are different, then there will be one intersection point.

 

It is important to also understand when a system has 0 solutions, or infinitely many solutions.

A "solution" is simply when the two lines share the same x,y components.

 

For a system to have 0 solutions, meaning the lines never touch each other, the system would be composed of two parallel lines that are separated. For instance y=x and y=x+1 will never touch.

 

For a system to have infinitely many solutions, then the two lines will be overlapping each other. That is for any x,y (infinite possibilities) of equation 1, there are an equal (infinite) number of overlaps with equation 2. 

An example: x + y = 1 , and 2x + 2y = 2. If you simply the second equation, you will see that it equals x + y = 1, and shares all of the same points as the first equation. 

 

-Joseph

Answer 2

There are several ways.

 

In this case, an easy way is to figure out the slope of the two lines that represent the equations.  If the slopes are different, then the lines intersect at one point, which means one solution.

 

In this case, the slopes are -1/2 and -4.

 

Therefore there is one solution.