a). The first step in solving by substitution is always solving for one of the variables. Since x in the second equation has coefficient
1
, we'll choose this variable to isolate.
x
−
2
y
=
−
8
→
x
=
2
y
−
8
We now substitute this into the first equation.
3
(
2
y
−
8
)
−
6
y
=
−
12
6
y
−
24
−
6
y
=
−
12
0
y
=
12
This is true for no real value of
y
, therefore this system has no real solution.
b). Let's do a little bit of work with the first equation.
3
x
−
6
y
=
−
12
We factor out a
3
.
3
(
x
−
2
y
)
=
−
12
Divide both sides by
3
x
−
2
y
=
−
4
We get an equation that is identical to the second on the left-hand side, but different on the right-hand side. What does this mean?
Suppose we were to graph both lines. We would first convert to slope-intercept form.
x
−
2
y
=
−
4
→
−
2
y
=
−
4
−
x
→
y
=
1
2
x
+
2
For the second equation:
x
−
2
y
=
−
8
→
−
2
y
=
−
8
−
x
→
y
=
1
2
x
+
4
These lines have equal slopes but different y-intercepts. This means that these are parallel lines, which is graphical proof that they will never intersect.
Hopefully this helps!