SOLUTION: Do the equations 4x−6y=10 and 6x−12y=15 graph as the same line?

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Question 1149922: Do the equations 4x−6y=10 and 6x−12y=15 graph as the same line?
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
I reworked that, don't know what happened to it.
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I found it.
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Do the equations 4x−6y=10 and 6x−12y=15 graph as the same line?
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4x−6y=10 ---> 2x-3y = 5
6x−12y=15 ---> 2x-4y = 5
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Different lines.
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Or solve each for y:
4x−6y=10 ---> y = (2/3)x - 5/3
6x−12y=15 --> y = (1/2)x - 5/4
Different slopes, different lines.
If the slopes are found to be equal, the y-intercepts must also be equal to be the same line.
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.

Write the coefficients and the right side value for the first and the second equations, separately.


For the first equation it is     (4,-6,10).

For the second equations it is   (6,-12,15).


Had these two triples be proportional  (with one common proportionality coefficient for all three terms), 

then two equations would present the same line.


But in your case it is not so.


So, these two equations represent DIFFERENT lines.

Answered and explained.

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See the lesson
    - Geometric interpretation of the linear system of two equations in two unknowns
in this site.

It is very important subject, and you should read about it from different sources to get full understanding (!).


Happy learning (!)


Come again to this forum soon to learn something new (!)



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