Question

Is this system of equations consistent, inconsistent, or dependent?
4x-3y=15
7y-35=9 1/3x

A. consistent
B. inconsistent
C. dependent
D. inconsistent and dependent
E. none of these

Answer 1

@DanJS can you help me? i'm leaning more towards D?

Answer 2

What are the definitions of each of those cases, that you know

Answer 3

Here is what i think:

Inconsistent = no solutions to the system, no point (x,y) that satisfies both equations. They are Parallel lines.

Dependent = There are infinitely many solutions to the system. This happens if both equations are the same line, they overlap at an infinite amount of points (x,y)

Consistent = the lines cross at one point, and there is a unique solution (x,y) that satisfies the equations.

Answer 4

That sound about good?

Answer 5

yes, those are basically the same definition to what i have written.

Answer 6

Just by those responses, without considering the equations, i think we can rule out D, i dont think it can be dependent and inconsistent at the same time can it?

Answer 7

And it has to be one of those 3 cases i think, i cant think of a system that is not one of the three, so that rules out E. None of the above

Answer 8

So there are Infinitely many solutions, No solutions, Or , 1 Unique Solution.

In other words, Parallel Lines, The Same Line Overlapping, or Lines that cross Once... in a 2-D space, we aren't considering Skew lines here

Answer 9

So, lets take a look at the equations now

Answer 10

4x - 3y = 15
7y - 35 = 9 1/3x

Answer 11

is that 9 and 1/3 x?

Answer 12

yes sorry i wasnt sure how to type that mixed fraction.

Answer 13

that is right, just wondering if it is a mixed number for sure

Answer 14

\[9 \frac{ 1 }{ 3 }x\]

Answer 15

The easiest way to tell which case you have , is to try and solve the system. There are other ways that are quicker , i can tell you after trying to solve them first

Answer 16

okay let me try and solve it quickly

Answer 17

turn that mixed number to an improper fraction first, then multiply the whole equation by the denominator to get rid of the fraction

Answer 18

7y - 35 = 9 1/3x

7y - (28/3)x = 35

21y - 28x = 105
----------------------- so the two equations w/o fractions are

4x - 3y = 15
-28x + 21y = 105

Answer 19

ok thats what i just got as well.

Answer 20

before solving, let me show you this neat trick

Answer 21

alright :)

Answer 22

well, actually no, try to solve first

Answer 23

okay one second.

Answer 24

i'm solving for x and y correct?

Answer 25

right, you could solve one for x or y, then substitute that into the second equation

Answer 26

4x - 3y = 15
-28x + 21y = 105

Answer 27

alright would x be 15/4 and y -5 ?

Answer 28

hmm, you must have gone wrong someplace... ill start going through, lets try to solve each one for y = mx + b first ok?

Answer 29

okay

Answer 30

4x - 3y = 15
3y = 4x - 15 ------------> y = (4/3)x - 15

-28x + 21y = 105
21y = 28x + 105 ----------> y = (4/3)x + 15

-------------------------------------------
The two equations are

\[y = \frac{ 4 }{ 3 }x - 15~~~and~~~y=\frac{ 4 }{ 3}x+15\]

Answer 31

Any part of that you don't see how to do?

Answer 32

okay yeah that's what i was getting, except \[\frac{ 4 }{ 3 }x-5\]

Answer 33

so that's where i got the -5. i see what you did though

Answer 34

ok, this is another way to find the solution,... now that you have both in the form y = mx+b

set y1 = y2 and solve for x , we do that because we are looking for a point (x,y) that will work in both equations. So Y in the first equation should equal Y in the second equation.

Y = Y, replace with mx+b = mx+b and solve for x

Answer 35

\[\frac{ 4 }{ 3}x + 15 = \frac{ 4 }{ 3 }x - 15\]

Answer 36

what do you get for x when you solve that?

Answer 37

lets see. what if i don't get anything?

Answer 38

take off 4/3 x on both sides, you are left with

15 = -15 right?

Answer 39

That is a false statement. This means that there is NO SOLUTION (x,y) that will work in both equations.

Answer 40

you're right it would bring it down to 15=-15

Answer 41

From the two lines in y = mx+b form...

y = (4/3)x + 15
y = (4/3)x - 15

what do you notice about the slopes (m), and the Y intercepts (b)

Answer 42

There are no solutions to the two equations. The solutions are where the two lines intersect (x,y). Here they are parallel, They have the same slope (4/3) but different y-intercpets +15 and -15

Answer 43

That is why when you tried to solve it, you got a nonsense answer like -15 = 15. That means there are no solutions, and the lines are parallel.

Answer 44

From the initial post i did with the definitions, Which one is this then?

Inconsistent = no solutions to the system, no point (x,y) that satisfies both equations. They are Parallel lines.

Dependent = There are infinitely many solutions to the system. This happens if both equations are the same line, they overlap at an infinite amount of points (x,y)

Consistent = the lines cross at one point, and there is a unique solution (x,y) that satisfies the equations.

Answer 45

I will show you a faster way to do these problems now if you would like.

Answer 46

so it would solely be inconsistent then.

Answer 47

yep, inconsistent = no solutions -15 = 15 FALSE

Answer 48

Remember it like, if you get a dumb answer like 10 = 25 , obviously false, then another word for that is it is inconsistent.

Answer 49

Here is a fast way to tell if lines are either Parallel, or the Same Line (Inconsistent or dependent).....

4x - 3y = 15
-28x + 21y = 105

Multiply the first equation by a constant number (-7), the system is then...

-28x + 21y = -105
-28x + 21y = 105

Notice by multiplying the first equation by a constant (this doesnt change the equation at all if you do it to every term), the left hand side becomes identical for the 2

Answer 50

If you can multiply an equation by a constant, and the equation becomes the same left hand side, the lines are parallel.

Answer 51

We say , lines are parallel if one is a scalar multiple, in this case (-7) of the other line.

Answer 52

If you multiply through , and both sides of the equation becomes the same, then obviously you have the same line and they are dependent.

For Example, Pretend the 15 in the first was a -15

4x - 3y = -15
-28x + 21y = 105

Now multiply the first equation again by (-7) and you get...
-28x + 21y = 105
-28x + 21y = 105

Same exact equation, same line, dependent, infinite solutions

Answer 53

that is just a quicker way to tell real fast if you have parallel lines, or the same line written in 2 different forms.

Answer 54

Here is a little page i found to review if needed, going over each type of system...
http://www.algebra.com/algebra/homework/coordinate/Types-of-systems-inconsistent-dependent-independent.lesson

Answer 55

if you have another one you want to run through for practice, tag me in