SOLUTION: Find the constant m for which all three lines x - y = -m …eq. 1 , 2x + y = m - 1 …eq. 2, and x + 5y = 4m + 1 …eq. 3 intersect at one point. Note: Can you please show your

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Question 1180819: Find the constant m for which all three lines x - y = -m …eq. 1 , 2x + y = m - 1 …eq. 2, and x + 5y = 4m + 1 …eq. 3 intersect at one point.
Note: Can you please show your full solution? Thank you!
Found 3 solutions by josgarithmetic, MathLover1, ikleyn:
Answer by josgarithmetic(39618)   (Show Source): You can put this solution on YOUR website!
Eq. 1        x-y=-m
Eq. 2       2x+y=m-1
Eq. 3        x+5y=4m+1








and continue to find y and x, using the found value of m.














reworked, fixed mistake
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

Find the constant m for which all three lines
…eq. 1
…eq. 2
…eq. 3
----------------------------------------
…eq. 1 , solve for
…eq. 1a
…eq. 2, substitute




........eq,2a
go to
…eq. 1a , substitute

..................eq.1b
go to
…eq. 3 , substitute and
...solve for





go to
........eq,2a , substitute



…eq. 1
…eq. 2
…eq. 3
----------------------------------------
…eq. 1
…eq. 2
…eq. 3
----------------------------------------
all three lines are intersecting at one point which is (, )




Answer by ikleyn(52797)   (Show Source): You can put this solution on YOUR website!
.
.
Find the constant m for which all three lines x - y = -m …eq. 1 , 2x + y = m - 1 …eq. 2, and x + 5y = 4m + 1 …eq. 3 intersect at one point.
Note: Can you please show your full solution? Thank you!
~~~~~~~~~~~~~~~~ 

You have these three starting equations

     x - y = -m        (1)

    2x + y =  m - 1    (2)

    x + 5y = 4m + 1    (3)



Add equations (1) and (2).  You will get

     3x    = -1;   hence  x = .


Next, from equation (1) express  y = x + m = .


Now, substitute both these expressions  x =   and  y =   into equation (3).  You will get

     +  = 4m + 1.    (4)



You just have one equation for one single unknown m in both sides.   It is easy to solve.

So, multiply equation (4) by 3 (both sides) to run from denominator.  You will get

    -1 + 15m - 5 = 12m + 3

         15m - 6 = 12m + 3

          3m     = 3 + 6 = 9

           m             = 9/3 = 3.


ANSWER.  m = 3.

Solved.

Quite simple and reasonably short.   Isn't it ?


The strategy was to construct equation for single unknown  "m"  and then solve it.


Also notice that as soon as you found  "m",  you can find  x  and  y  momentarily 

    x =    (you just know it);   y =  =  = 2  = .


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Do not forget to post your  "THANKS"  to me for my teaching.


\\\\\\\\\\\\\\


Ignore the post by @josgarithmetic,  since his solution is  INCORRECT.



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