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Start with the first equation.
Subtract
from both sides.
Subtract
from both sides.
Combine like terms on the right side.
Divide both sides by
to isolate
.
Simplify.
-----------------------
Now move onto the given equation.
Subtract
from both sides.
Add
to both sides.
Combine like terms on the right side.
Divide both sides by
to isolate
.
Simplify.
So when we graph the two equations, we get
Graph of
(red) and
(green)
Notice how the slope of the two equations is
. So the perpendicular slope is
. Now simply draw any perpendicular line to the equations. I'm going to graph
Graph of
(red) ,
(green), and the perpendicular line
(blue)
Now use any method to find the two intersections. I used a graphing utility to get the two intersections at the points
)
and
)
Now let's find the distance between the two points. This distance will be the same as the shortest distance between the two lines
Start with the distance formula.
Plug in
,
,
, and
. These are the coordinates of the intersections.
Subtract
from
to get
.
Subtract
from
to get
.
Square
to get
.
Square
to get
.
Add
to
to get
.
Take the square root of
to get
.
So our answer is
So the distance between the two points is 3 units.
So this means that the shortest distance between the two lines is also 3 units.
So the answer choice is A)
(