Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 01 (c)

MATH152 April 2017

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[hide] Question B 01 (c)

Three engineering students, Xiuying, Arjan, and Marianne decide to take a summer job painting homes. Xiuying paints three times as fast as Marianne and twice as fast as Arjan and Marianne combined. All three together paint a room in four hours. (c) Solve the system above using Gaussian elimination on the augmented matrix. How many rooms can each of the three friends paint in an hour? Check that your answer matches the original information in the question.

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!

[show] Hint
Write the equations in terms of 3 variables, say $X,M,A$ and then use the coefficients to write the augmented matrix. Remember the information given is about the Rate and at which the students paint and their relative ratios.

Checking a solution serves two purposes: helping you if, after having used the hint, 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

From (b) the augmented matrix is: ${\begin{bmatrix}1&0&-3&|0\\1&-2&-2&|0\\1&1&1&|0.25\\\end{bmatrix}}$ Then we follow the steps for Gaussian elimination: Multiply the 1st row by $-1$ and add it to the 2nd and 3rd row: ${\begin{bmatrix}1&0&-3&|0\\0&-2&1&|0\\0&1&4&|0.25\\\end{bmatrix}}$ Then switch the 2nd and 3rd rows with each other ${\begin{bmatrix}1&0&-3&|0\\0&1&4&|0.25\\0&-2&1&|0\\\end{bmatrix}}$ Then multiply the 2nd row by $2$ and add it to 3rd row: ${\begin{bmatrix}1&0&-3&|0\\0&1&4&|0.25\\0&0&9&|0.5\\\end{bmatrix}}$ Divide the 3rd by 9: ${\begin{bmatrix}1&0&-3&|0\\0&1&4&|0.25\\0&0&1&|1/18\\\end{bmatrix}}$ Last step is to cancel values above 1 in the 3 third column, therefore the final form of the augmented matrix in upper echeleon form is: ${\begin{bmatrix}1&0&0&|1/6\\0&1&0&|1/36\\0&0&1&|1/18\\\end{bmatrix}}$ Answer: $\color {blue}x_{1}={\frac {1}{6}}Room/Hour,\,x_{3}={\frac {1}{18}}Room/Hour,\,x_{2}={\frac {1}{36}}Room/Hour$

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