Question

how many solutions does the system of equations have?
2x= -10+6 and x+5y=3
one
two
infinitely many
none

how many solutions does the system of equations have?
y=5x+7and3y-15x=18
one
two
infinitely many
none

how many solutions does the system of equations have?
x-4y=12and5x-20y=60
one
two
infinitely many
none

how many solutions does the system of equations have?
y-5x=-6 and 3y-15x= -12
one
two
infinitely many
none

what is the solutions of the system?
4x-y= -3
6x-y= -8

(5,5.5)
(-2.5,-7)
(-2.5,2.2)
(2.5,-7)


What is the solution of the system?
5x+6y=20
9x+11y=32

(28, -20)
(28,20)
(-28, 20)
(20, -28)


please help. Please and thank you!

Answer 1

assuming you meant
2x= -10y+6
x+5y=3
then multiply the 2nd by 2 and you then have
2x = -10y + 6
2x + 10y = 6
you can see that they are the same equation, so there are infinitely many solutions. That is, any point on the first line is also on the second, since they are the same line.

If you do this same kind of manipulation and you get something like
x+2y = 4
x+2y = -3
then it is clear that no number can satisfy both equations. The graphs are parallel lines.

Finally, if you check the slopes and they are different, then clearly the lines must intersect in exactly one point.
recall that Ax+By=C has slope m = -A/B