Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 1 (c)

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MATH152 April 2015

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Question B 1 (c)
Uno, Duo and Traea are three friends. They all owe money to a loan shark. All together they owe $600. Duo owes $200 more than Uno. Uno and Duo combined owe as much as Traea. (c) Solve the system in part (a) using Gaussian elimination on the augmented matrix. How much does each person owe?

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Apply Gaussian elimination: reduce the matrix to reduced row echelon form (wherein all the pivots are 1, and each pivot is the only nonzero entry in its column).

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Solution

We perform Gaussian elimination to compute the reduced row echelon form of the augmented matrix: ${\begin{aligned}\left[{\begin{array}{ccc|c}1&1&1&600\\-1&1&0&200\\1&1&-1&0\end{array}}\right]&\xrightarrow {R_{3}-R_{1}} \left[{\begin{array}{ccc|c}1&1&1&600\\-1&1&0&200\\0&0&-2&-600\end{array}}\right]&\xrightarrow {R_{2}+R_{1}} \left[{\begin{array}{ccc|c}1&1&1&600\\0&2&1&800\\0&0&-2&-600\end{array}}\right]&\xrightarrow {{\frac {R_{2}}{2}},-{\frac {R_{3}}{2}}} \left[{\begin{array}{ccc|c}1&1&1&600\\0&1&1/2&400\\0&0&1&300\end{array}}\right]\\&\xrightarrow {R_{1}-R_{2}} \left[{\begin{array}{ccc|c}1&0&1/2&200\\0&1&1/2&400\\0&0&1&300\end{array}}\right]&\xrightarrow {R_{2}-{\frac {1}{2}}R_{3}} \left[{\begin{array}{ccc|c}1&0&1/2&200\\0&1&0&250\\0&0&1&300\end{array}}\right]&\xrightarrow {R_{1}-{\frac {1}{2}}R_{3}} \left[{\begin{array}{ccc|c}1&0&0&50\\0&1&0&250\\0&0&1&300\end{array}}\right].\end{aligned}}$ Thus ${\textstyle \color {blue}x_{1}=50,x_{2}=250,x_{3}=300.}$

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