Question 1180819
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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!
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<pre>
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 = {{{-1/3}}}.


Next, from equation (1) express  y = x + m = {{{m - 1/3}}}.


Now, substitute both these expressions  x = {{{-1/3}}}  and  y = {{{m-1/3}}}  into equation (3).  You will get

    {{{-1/3}}} + {{{5*(m-1/3)}}} = 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.


<U>ANSWER</U>.  m = 3.
</pre>

Solved.


Quite simple and reasonably short.  &nbsp;&nbsp;Isn't it ?



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



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


<pre>
    x = {{{-1/3}}}   (you just know it);   y = {{{m-1/3}}} = {{{3 - 1/3}}} = 2 {{{2/3}}} = {{{8/3}}}.
</pre>


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