Question

When solving a system of equatios using the additio or substitution method, how can you tell if a system has infinitely many or no solutions? What's the relationship between the graph in these situations? Show all work neatly.

Answer 1

when solving system of equations using addition or substitution method

if you get the variables cancelled out and left hand side equal to right hand side it means system has infinite many solutions

for example

2x + 4y = 8

x + 2y = 4

multiplying equation 2 by -2 and adding equation 1 to it

-2x - 4y = -8

2x + 4y = 8

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

0 = 0

since all the terms cancel out and we get 0 = 0

this means this system has infinite many solutions

but if on solving the system the variables get cancelled and left hand side is not equal to right hand side

then system has no solutions

for example

2x + 4y = 8

x + 2y = 9

multiplying equation 2 by - 2 and adding equation 1 to it

- 2x - 4y = -18

2x + 4y = 8

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

0 = -10

left hand side is not equal to right hand side

hence, this system has no solutions

the graphs of infinite solutions are two lines that overlap

the graphs of no solution are two parallel lines as parallel lines never meet . hence no solution