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Determine, in accordance to $k$, how many solutions does the given system of equations have: $$ \begin{cases}kx+(k+1)y=k-1\\4x+(k+4)y=k\end{cases} $$ And check, for which values of $k$ this system has exactly one solution lying within the boundaries of a triangle of vertexes: $A=(0, 0), B=(\frac{2}{3}, 0), C=(0, 2)$.

Can I just solve it using determinants? I mean: $$D=k(k+4)-4(k+1)=k^2-4=(k+2)(k-2)$$ $$D_x=(k-1)(k+4)-(k+1)k=2k-4$$ $$D_y=k^2-4(k-1)=k^2-4k+4$$

Our system has one solution if $D$ is not $0$. This happens when $k$ is neither $2$ nor $-2$.

It has no solutions when $D=0$ and ($D_x \neq 0$ or $D_y \neq 0$). $D=0$ for $k=2$ or $k=-2$, $D_x \neq 0$ when $k \neq 2$ and $D_y \neq 0$ when $k \neq 2$. Though for $k=2$ our $D$ equals $0$, neither $D_x$ nor $D_y$ is $\neq 0$ so it has no solutions only when $k=-2$.

It has infinitely many solutions for $D=0$ (which means $k=2$ or $k=-2$), $D_x = D_y = 0$. This happens when $k=2$.

Summing up: $$ \begin{cases} \text{no solution}, &\text{for } k=-2, \\ \text{infinitely many solutions}, &\text{for } k=2, \\ \text{one solution}, &\text{for every other } k. \end{cases} $$

Is this all right? I need to know that before proceeding to the triangle-thing.

Srivatsan
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Jameson
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    It's nice to see your work (and that too, very detailed). Currently I ran out of my votes; I will upvote this question when I get them back. – Srivatsan Oct 24 '11 at 14:56
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    I only wish people asking for help here were like you... for now, have my upvote. – J. M. ain't a mathematician Oct 24 '11 at 15:06
  • Oh, thank you very much :) But regarding my question: turns out the triangle part is not as easy as I suspected. I mean, firstly, I thought it's only left to find such x'es and y'es that $x\in (0,\frac{2}{3})$ and $y\in (0, 2)$ but come to think of it, it's not true. We might end up with x=1.99 and y=0.65. Then, though it fulfills our requirements, it's not within the boundaries of the triangle. How should I approach it, then? – Jameson Oct 24 '11 at 15:13

2 Answers2

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First, let's get back to the mainstream terminology: We have a linear system $Ax=b$ with the data $$ \pmatrix{k &k+1\\4 &k+4}\pmatrix{x\\y}=\pmatrix{k-1\\k} $$ Now if $A$ is invertible then there exists a unique solution which is given by $$ \pmatrix{x\\y} = \pmatrix{k &k+1\\4 &k+4}^{-1}\pmatrix{k-1\\k} = \frac{1}{k^2-4}\pmatrix{k+4 &-(k+1)\\-4&k}\pmatrix{k-1\\k} = \pmatrix{\frac{2}{k+2}\\ 1-\frac{4}{k+2}} $$ As you have computed, the easiest way to conclude invertibility is having the determinant nonzero. Computing the determinant gives: $\det A = k^2-4 = (k-2)(k+2)$, Thus, we have a unique solution for all $|k|\neq 2$.

Regarding the zero determinant cases, let's look at the system at those points:

($k=2$) $$ \pmatrix{2&3\\4&6}\pmatrix{x\\y} = \pmatrix{1\\2} $$ We can clearly see that the second equation is the 2-multiple of the first one. Hence we actually only have $2x+3y = 1$

($k=-2$) $$ \pmatrix{-2&-1\\4&2}\pmatrix{x\\y} = \pmatrix{-3\\-2} $$ We can see that the right hand side is not in the range of $A$, hence there is no solution. Another way to look at it is rewriting as $$ \pmatrix{-2\\4}x+\pmatrix{-1\\2}y = \pmatrix{-1\\2}(2x+y) = \pmatrix{-1\\2}\alpha = \pmatrix{-3\\-2} $$ It is obvious that no alpha can satisfy that equality hence no solution.

Regarding the triangle question, you can rewrite the constraints as $$ \pmatrix{-3 &-1\\1 &0\\0&1}\pmatrix{x\\y}\geq \pmatrix{-2\\0\\0} $$ Now plug in the $k$ dependent solution into this and we get $$ \pmatrix{1-\frac{2}{(k + 2)}\\ \frac{2}{(k + 2)}\\ 1 - \frac{4}{(k + 2)}}\geq 0 $$ This gives us the condition on $k$: $$ 0\leq \frac{2}{k+2} \leq \frac{1}{2} \implies k>2 $$ Note that we ruled out $k=2$ case since there are infinitely many solutions in that case violating the requirement given in the question.

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I don't know your terminology, but the answer is correct. For $k=2$ the equations are multiples of each other. For $k=-2$ the left sides are multiples but the rights do not respect that, so the equations are inconsistent.

For the triangle part, in the case there is only one solution, you should be able to express the point as a function of $k$. In other words, the solution will be $(f(k),g(k))$ This will be a continuous function of $k$, so you can find the values of $k$ where it hits the edges of the triangle. For $k=2$ you have a line of solutions-does it hit the triangle? It can only have a single solution in the triangle if it hits one of the vertices and you consider that "within" the triangle.

Ross Millikan
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  • Thank you. Isn't there any easier option of expressing the points within the triangle? I mean, this solution with functions seems quite complicated to me, as opposed to the rest of the problem... – Jameson Oct 24 '11 at 15:26
  • $f$ and $g$ will come out from solving the simultaneous equations. I would multiply the first by $4$, the second by $k$, and subtract. Solving the result for $y$ will get $g(k)=\frac{4(k-1)-k^2}{4(k+1)-k(k+4)}$ which I suspect can be simplified. – Ross Millikan Oct 24 '11 at 15:35
  • I mean, I know that $x=\frac{D_x}{D}$ and $y=\frac{D_y}{D}$ but I don't know what to do with it next. We come to $x=\frac{2}{k+2}$ and $y=\frac{k+2}{k-2}$ but what did this give me? – Jameson Oct 24 '11 at 15:38
  • Since $y\in(0,2), k>6$ and we are looking for the intersection of this curve with the line $y=2-3x$, which happens at $k=10, x=\frac{1}{6}, y=\frac{3}{2}$. It will stay within the triangle for all $k>10$, tending toward $(0,1)$ as $k \to \infty$ – Ross Millikan Oct 24 '11 at 16:08
  • How do you know that k>6 when $y\in(0,2)$? Also, why are you looking for the intersection with $y=2-3x$? – Jameson Oct 24 '11 at 16:10
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    I just solved $2=\frac{k+2}{k-2}$ and observed it decreases from there as $k$ gets larger. I missed the branch where $k<-2$, but on that one $x<0$ and the point is always outside the triangle. Then I observed that the point moves toward the triangle and will enter it by crossing that side (which is the one from (0,2) to (2/3,0)) – Ross Millikan Oct 24 '11 at 16:18
  • OK, I know where did the y equation come from, sorry :) But still - why k>6? – Jameson Oct 24 '11 at 16:20
  • OK, so I understand that. However, why were you looking for the intersection with y=2-3x? I know where the equation comes from, it's the third (non-right) edge of the triangle - but why the intersection? – Jameson Oct 24 '11 at 16:25
  • At $k=6$, the point is (1/4,2) which is outside the triangle. I just made a table in Excel of k,x,y and found it was moving down and to the right as $k$ increased, so it will enter the triangle by crossing that side. Solving that equation finds the value of $k$ where it crosses. – Ross Millikan Oct 24 '11 at 16:33
  • Oh, I see. And from what I see, it doesn't go past the triangle for any k greater than 10, right? I mean - all the k's greater than 10 are the answers? – Jameson Oct 24 '11 at 16:36
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    Right you are. That was the point that it approaches (0,1) which is on the edge of the triangle. – Ross Millikan Oct 24 '11 at 16:43
  • Whew! Thank you for your patience and clarification, sir :D – Jameson Oct 24 '11 at 16:44
  • Oh, I believe you've made a mistake. You solved $y=\frac{k+2}{k-2}$ which is untrue since $y=\frac{(k-2)^2}{(k+2)(k-2)}=\frac{k-2}{k+2}$. The solution isn't right, then, right? – Jameson Oct 24 '11 at 17:30
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    I took it from your previous comment, but yes I see $y=\frac{k-2}{k+2}$. Now we want $k>2$ for $y$ and $k>1$ for $x$. So it enters the triangle at $k=2, (1/2,0)$ and continues toward (0,1) never leaving the triangle. This now agrees with percusse – Ross Millikan Oct 24 '11 at 17:49
  • Right. Thank you very much :) – Jameson Oct 24 '11 at 19:55
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    Glad to help somebody that is thinking about the problem. – Ross Millikan Oct 24 '11 at 19:58