give examples of a system of equations with infinitely many solutions.

Question

Give an example of a system of equations with infinitely many solutions. 

Answer 1

If by a system of equations you want two "different" equations with infinitely many points (solutions) in common you could take any linear equation like the one Hamilton gave you y = x + 1 and multiply it by 2 to get 2y = 2x + 2. 

 

If you try to solve you will get 0 = 0 because they are the same line and every (x,y) point satisfying one equation satisfies the other.

Answer 2

x-2y=3

-3x+6y=-9

 

one line is sitting on top of the other line, therefore you have infinitely many solutions

in other words, the two equations are equal to each other

to show this...

x-2y=3

multiply both sides by -3

you get -3x+6y=-9 which is the second equation, or...

multiply both sides by 3

you get 3x-6y=9

add this to -3x+6y=-9

3x-6y=9

-3x+6y=-9

adding...

0+0=0

0=0

you get a true statement when you solve

if two lines intersect in an "X" shaped pattern, there is one solution

for example...

x+2y=8

5x+3y=19

the solution is (2,3)

 

if the lines are parallel there is no solution

for example...

x-y=5

x-y=7

the lines have the same slope but different y-intercepts so they will never intersect

if you tried to solve this system you would get a false statement

 

from the first example...

x-2y=3

2y=x-3

y=(1/2)x-(3/2)

x=0, y=-3/2

x=1, y=-1

x=-1, y=-2

and so on

Answer 3

 

A system with 0 solutions might look something like:

x + y = 1

x + y = 2

There are no choices of x and y that make both of those inequalities work, so no solutions.

 

A system with 1 solution might look like:

x = 3

You could also have a system of two non-parallel lines (since they must intersect at exactly one point).

 

A system with infinitely many solutions can be as simple as a line:

y = x + 1

There are infinitely many choices for x, and y comes along for the ride. (1, 2), (2, 3), (3, 4), ... are all solutions.