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

Algebra ->  Points-lines-and-rays -> 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      Log On


   

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) About Me  (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

system%28x-y=-m%2C2x%2By=m-1%2Cx%2B5y=4m%2B1%29

system%28x-y%2Bm=0%2C2x%2By-m=-1%2Cx%2B5y-4m=1%29

system%28x-y%2Bm=0%2C0%2B3y-3m=-1%2C0%2B6y-5m=1%29

system%28x-y%2Bm=0%2C3y-3m=-1%2C0%2Bm=3%29
and continue to find y and x, using the found value of m.














reworked, fixed mistake

Answer by MathLover1(20850) About Me  (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+%2B+y+=+m+-+1+…eq. 2
x+%2B+5y+=+4m+%2B+1 …eq. 3
----------------------------------------
x+-+y+=+-m …eq. 1 , solve for x
x++=+y-m …eq. 1a
+2x+%2B+y+=+m+-+1+…eq. 2, substitute x
+2%28y-m%29+%2B+y+=+m+-+1+
+2y-2m+%2B+y+=+m+-+1+
+3y+=+2m%2Bm+-+1+
+3y+=+3m+-+1+
+y+=+m+-+1%2F3+........eq,2a
go to
x++=+y-m …eq. 1a , substitute y
x++=+m+-+1%2F3+-m
x++=+-+1%2F3+ ..................eq.1b
go to
x+%2B+5y+=+4m+%2B+1 …eq. 3 , substitute x and y
-1%2F3+%2B+5%28m+-+1%2F3%29+=+4m+%2B+1...solve for m

-1%2F3+%2B+5m+-+5%2F3+=+4m+%2B+1
+5m+-+2+=+4m+%2B+1
+5m+-+4m+=+2+%2B+1
+m+=+3
go to
+y+=+m+-+1%2F3+........eq,2a , substitute m
+y+=+3+-+1%2F3+
+y+=+8%2F3+

x+-+y+=+-3 …eq. 1
+2x+%2B+y+=+3+-+1+…eq. 2
x+%2B+5y+=+4%2A3+%2B+1 …eq. 3
----------------------------------------
x+-+y+=+-3 …eq. 1
+2x+%2B+y+=+2+…eq. 2
x+%2B+5y+=+13 …eq. 3
----------------------------------------
all three lines are intersecting at one point which is (-1%2F3, +8%2F3)




Answer by ikleyn(52790) About Me  (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 = -1%2F3.


Next, from equation (1) express  y = x + m = m+-+1%2F3.


Now, substitute both these expressions  x = -1%2F3  and  y = m-1%2F3  into equation (3).  You will get

    -1%2F3 + 5%2A%28m-1%2F3%29 = 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 = -1%2F3   (you just know it);   y = m-1%2F3 = 3+-+1%2F3 = 2 2%2F3 = 8%2F3.


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