Answer:
Choice d.
Step-by-step explanation:
Consistent means the linear functions will have at least one solution.
Independent means the it will just that one solution.
Dependent means the system will have infinitely many solutions.
So we are looking for a pair of equations that consist of the same line.
y=mx+b is slope-intercept form where m is the slope and b is the y-intercept.
If m and b are the same amongst the pair, then that pair is the same line and the system is consistent and dependent.
If m is the same and b is different amongst the pair, then they are parallel and the system will be inconsistent (have no solution).
If m is different, then the pair will have one solution and will be consistent and independent.
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Choice A:
First line has m=1 and b=4.
Second line has m=1 and b=-4.
These lines are parallel.
This system is inconsistent.
Choice B:
First line is x+y=4 and it isn't in y=mx+b form.
Second line is 2x+2y=6 and it isn't in y=mx+b form.
These lines do have the same form though. This is actually standard form.
If you divide second equation by 2 on both sides you get: x+y=3
The lines are not the same.
We are looking for the same line.
Now we could go ahead and determine to call this system.
If x+y has value 4 then how could x+y have value 3. It is not possible. There is no solution. These lines are parallel.
Need more convincing. Let's put them into slope-intercept form.
x+y=4
Subtract x on both sides:
y=-x+4
m=-1 and b=4
x+y=3
Subtract x on both sides:
y=-x+3
m=-1 and b=3.
The system is inconsistent because they are parallel.
Choice c:
I'm going to go ahead in put them both in slope-intercept form:
3x+y=3
Subtract 3x on both sides:
y=-3x+3
So m=-3 and b=3
2y=6x+6
Divide both sides by 2:
y=3x+3
So m=3 and b=3
The m's are different so this system will by consistent and independent.
Choice d:
The goal is the same. Put them in slope-intercept form.
4x-2y=6
Divide both sides by -2:
-2x+y=-3
Add 2x on both sides:
y=2x-3
m=2 and b=-3
6x-3y=9
Divide both sides by -3:
-2x+y=-3
Add 2x on btoh sides:
y=2x-3
m=2 and b=-3
These are the same line because they have the same m and the same b.
This system is consistent and dependent.
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