Science:Math Exam Resources/Courses/MATH221/April 2013/Question 01

MATH221 April 2013

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[hide] Question 01
Consider the system of linear equations: ${\begin{aligned}3x&\ +&4y&\ +&7z&\ =&3\\-9x&\ +&6y&\ +&9z&\ =&3\\45x&\ -&12y&\ +&hz&\ =&k\end{aligned}}$ For which values of h and k does the system have: a) no solutions? b) exactly one solution? c) infinitely many solutions?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
Writing this system in matrix form, what will tell about the number of solutions?

[show] Hint 2
Pick carefully which row manipulations you will do, they will simplify your calculations quite a bit.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

[show] Solution

We can write this system in matrix form: $\left[{\begin{array}{ccc}3&4&7\\-9&6&9\\45&-12&h\end{array}}\right]\left[{\begin{array}{c}x\\y\\z\end{array}}\right]=\left[{\begin{array}{c}3\\3\\k\end{array}}\right]$ To know the number of solutions, we want to look at the row reduced form of the augmented matrix. First, perform the operations L3 → L3+5L2 and L2 → L2-3L1 $\left[{\begin{array}{cccc}3&4&7&3\\-9&6&9&3\\45&-12&h&k\end{array}}\right]\sim \left[{\begin{array}{cccc}3&4&7&3\\0&18&30&12\\0&18&45+h&15+k\end{array}}\right]$ Then, perform L3 → L3 - L2 $\sim \left[{\begin{array}{cccc}3&4&7&3\\0&18&30&12\\0&0&15+h&3+k\end{array}}\right]$ Noting that the first two rows of the matrix give a matrix of rank at least 2, the number of solutions depend only on the last row. We read the last row of this matrix and we obtain the equation $\displaystyle (15+h)z=3+k$ We can now read off the solutions from this equation base on whether or not $15+h$ and $3+k$ are equal to zero. a) No solution if $15+h=0,3+k\neq 0$ b) One solution if $15+h\neq 0$ c) Infinite solution if $\displaystyle 15+h=0,3+k=0$ Notice that this convers all possible values for $\displaystyle 15+h$ and $\displaystyle 3+k$.