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These are linear equations in two variables If they have a common solution, then they will cross each other at a point and hence will be consistent. We put them in the form ax + by + c = 0 4x - y - 15 = 0 and x + 5y + 12 = 0 Now If a1/a2≠b1/b2 then they will intersect at one point and will have a unique and consistent solution Here a1/a2 = 4/1 and b1/b2 = -1/5 Since a1/a2 is not equal to b1/b2, therefore it will have a unique solution and they are consistent.
Also since they share only one common point, they are consistent and independent
For your info, if \[a1/a2 \neq b1/b2\] then they are consistent and independent as they will cut each other at one point if \[a1/a2 = b1/b2 = c1/c2\] then they r consistent a dependent as the two lines will lie on each other if \[a1/a2 = b1/b2 \neq c1/c2\] then they r independent as they r parallel lines and will not meet/cut each other at even one point Hope this clears everything....
im more confused
because i have no idea how to do this
the other guy is telling me one thing and someone else told me another
First you have to put the given equations in ax + by + c = 0 form See how I did it.... Is that clear or not??
ok so 4x-y+15=0?
No it will be - 15 becoz when you shift a term from one side to the other, its sign changes
So, it will be 4x -y - 15 = 0
ok and x+5y-12=0 for the second equation?
No, x + 5y + 12 = 0 as sign on - 12 will become + when it is shifted to the other side.
ok then what
Now we check what is the ratio of the coefficients of x in the two equations. Here a1/a2 = 4/1 as in first eqn it is 4x and in the second eqn it is 1
Then we check for the ratio of the coefficients of the y in the two equations Here b1/b2 = -1/5 as in first eqn it is -y and in second eqn it is 5y
Finally we check for the ratio of the constant terms in the two equations here c1/c2 = -15/12
Clear till now??
nope lost me when you started talking about ratio's 5y
U u/sttod a1/a2 = 4/1 ??
yes it is 4x
I guess it wud be easier if cud talk, do u hv a SKYPE account???
I hv a SKYPE id hakki.singh
well it wud hv been very easy if we cud talk.....
i dont remember any of this from the hmwk
Maybe u are being taught a different method, mine is very simple and easy to follow if u get the hang once
x=0 then y=
No, first u hv to look at the coefficients of the x terms of both equns In the first one 4 is the coefficient of the term 4x In the second one 1 is the coefficient of the term x
So, a1/a2 ie the ratio of the coefficients of the x terms of the first eqn and the second eqn is 4/1
Now we go the y terms
In the first eqn it is -y and in the second eqn it is 5y so b1 is -1 abd b2 is 5 so b1/b2 is -1/5
Finally we look at the constant terms Here it is -15 in the first eqn and 12 in the second eqn So, c1/c2 is -15/12
Now we have to understand the terms consistent/inconsistent and dependent/independent
consistent means that the two eqns share at least one or more solutions This means that if their graphs cut each other at one point or the two lines coincide in which case they share infinite solutions inconsistent means that they do not share even one solution i.e. the line never meet
And in case they r consistent i.e they share one or more solutions, then they will be independent if they share just one solution and dependent if they share many solutions ie.e the line coincide
So we can have consistent and dependent consistent and independent and inconsistent
it is only one answer and the solution of the system typed in an ordered pair,
I'm coming to the solution, but is the above classifications as consistent and dependent, consistent and independent and inconsistent clear??
Fianlly, if a1/a2≠b1/b2 then they are consistent and independent as they will cut each other at one point if a1/a2=b1/b2=c1/c2 then they r consistent a dependent as the two lines will lie on each other if a1/a2=b1/b2≠c1/c2 then they r inconsistent as they r parallel lines and will not meet/cut each other at even one point
In your case a1/a2 = 4/1 and b1/b2 = -1/5 So, a1/a2 is not equal to b1/b2, so your pair of equations is consistent and independent
So, it is now clear tht yr system of pair of eqns is consistent and independent. To draw their graphs, you modify the first equation by separating x and y as follows 4x - 15 = y Now put various values like -1 , 0, 1, 4 for x and getting corresponding values of y Understood??
Similarly modify the second eqn as follows: 5y = -x -12 (remember signs change when you shift their side) so, -x -12 y = ----------- 5 Now, put various values for x and calculate corresponding values of y. Now plot the graphs for the two eqns separately and you will find them cutting each other at one point.
Your intersection point shud b (3, -3)
Hope now everything is clear??
ok and it is consistent and independent
yes it is
ok ty so much for all your help I really appreciate it
U r welcome. In this we have to be clear what consistent/inconsistent and dependent/independent mean. Then using the three ratio checks we can find whether there will be one solution, many solutions or no solution. Do I get a medal for this??