The coordinates below represent two linear equations.
how many solutions does this system of equations have?
> .<
line 1
x y
–6 –6
–4 –3
line 2
x y
0 3
2 6
a.
0
b.
exactly 1
c.
exactly 2
d.
infinitely many
the coordinates below represent two linear equations. how many solutions does this system of equations have?
line 1
x y
–4 8
4 6
line 2
x y
–1 1
3 5
a.
0
b.
exactly 1
c.
exactly 2
d.
infinitely many

Look at the first line in the first problem.
x    y
–6 –6
–4 –3
find the slope:
y2-y1 / x2-x1
-3+6 / -4+6
3 / 2
the slope is 3/2. now plug this into point-slope form, along with one of the ordered pairs. i'll use (-4,-3) because the numbers are smaller.
y-y1=m(x-x1)
y+3=3/2(x+4)
y+3=3/2x+6
y=3/2x+3

now do the same for the other line.
x y
0 3
2 6
actually, this one's convenient, because it tells you the y-intercept is 3 (when an ordered pair has a 0 as its x coordinate, that means its on the y axis). so we just have to find the slope.
y2-y1 / x2-x1
6-3 / 2-0
3/2
the line is y=3/2x+3. it's the same line as before. that means there are infinitely many solutions, because there are an infinite amount of points where the lines "cross" each other (since they lie on top of each other).

follow the same procedure for the next problem. if the equations turn out to be the same, it has infinite solutions. if the two lines have the same slope but different intercepts, that means they're parallel, and have no solution since the lines will never intersect. if the lines have different slopes, they will have one solution. and it makes no sense for lines to intersect at two different points, so you can ignore that option altogether.

hope this helps