Question 597567
Hi, there--
.
Here is the graph of the lines. Read below the graph for the rest of the solution.
.
{{{
     graph( 600, 400, -10, 10, -10, 10, 2, 6, 2x-2, 2x-6 )
}}}

a)red line: y=2
b)green line: y=6
c)blue line: y=2x-2
d)purple line: y=2x-6
.
The four lines do form a parallelogram. We know that it is a parallelogram because the opposite sides are formed by lines with the same slope: y=2 and y=6 both have a slope of 0, while y=2x-2 and y=2x-6 both have a slope of 2.
.
To find the vertices of the parallelogram, find the solution for the two equations that intersect at that point. I'll use the substitution method.
.
Find the red-blue vertex:
{{{y=2}}}
{{{y=2x-2}}}
.
Substitute 2 for y in the second equation and solve for x.
{{{2=2x-2}}}
{{{2x=4}}}
{{{x=2}}}
.
The red-blue vertex is (2,2).
.
Find the red-purple vertex:
{{{y=2}}}
{{{y=2x-6}}}
.
Substitute 2 for y in the second equation and solve for x.
{{{2=2x-6}}}
{{{2x=8}}}
{{{x=4}}}
.
The red-purple vertex is (4,2).
.
Find the green-blue vertex:
{{{y=6}}}
{{{y=2x-2}}}
.
Substitute 6 for y in the second equation and solve for x.
{{{6=2x-2}}}
{{{2x=8}}}
{{{x=4}}}
.
The green-blue vertex is (4,6).
.
Find the green-purple vertex:
{{{y=6}}}
{{{y=2x-6}}}
.
Substitute 6 for y in the second equation and solve for x.
{{{6=2x-6}}}
{{{2x=12}}}
{{{x=6}}}
.
The green-purple vertex is (6,6).
.