Science:Math Exam Resources/Courses/MATH221/December 2011/Question 01 (a)

MATH221 December 2011

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Question 01 (a)
Consider the system of equations in the variables x1, x2, x3: ${\begin{aligned}&x_{1}+2x_{2}+3x_{3}&=1\\&x_{1}+x_{2}+2x_{3}&=2\\&2x_{1}+3x_{2}+(5+t)x_{3}&=2\end{aligned}}$ Determine all the values of t for which the system is consistent.

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.

Hint 1
Writing this system in matrix form, what will tell you whether the system is consistent or not?

Hint 2
A linear system of three equations is consistent if it has a solution (x1, x2, x3).

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.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We can write this system in matrix form: $\left[{\begin{array}{ccc}1&2&3\\1&1&2\\2&3&5+t\end{array}}\right]\left[{\begin{array}{c}x_{1}\\x_{2}\\x_{3}\end{array}}\right]=\left[{\begin{array}{c}1\\2\\2\end{array}}\right]$ To know if the system is consistent, we want to look at the row reduced form of the augmented matrix. First, perform the operations L2 → L2-L1 and L3 → L3-2L1 $\left[{\begin{array}{cccc}1&2&3&1\\1&1&2&2\\2&3&5+t&2\end{array}}\right]\sim \left[{\begin{array}{cccc}1&2&3&1\\0&-1&-1&1\\0&-1&-1+t&0\end{array}}\right]$ Then, perform L2 → (-1)L2 $\sim \left[{\begin{array}{cccc}1&2&3&1\\0&1&1&-1\\0&-1&-1+t&0\end{array}}\right]$ Perform L3 → L3+L2 $\sim \left[{\begin{array}{cccc}1&2&3&1\\0&1&1&-1\\0&0&t&-1\end{array}}\right]$ If we read the last row of this matrix, we obtain the equation $\displaystyle tx_{3}=-1$ The system is inconsistent if t = 0 since that would give us the equation 0 = -1. Hence the system is consistent for all non-zero values of t.