Science:Math Exam Resources/Courses/MATH152/April 2011/Question B 01 (b)

MATH152 April 2011

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[hide] Question B 01 (b)

Three identical fields of size one hectare are planted with different amounts of three kinds of wheat (types 1, 2 and 3). The first field is divided in three equal parts with one type of wheat planted on each part. It yields 12 tons of total harvest. The second field is divided in two equal parts with type 2 planted on one part and type 3 planted on the other. It yields 10 tons. The third field is divided in two equal parts with type 1 planted on one part and type 2 on the other. It yields 16 tons. Write the system you found above in augmented matrix form.

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
Recall that an augmented matrix A with a vector b is $[A|\mathbf {b} ].$ This notation refers to the original matrix A with an extra column (in this case b) appended on.

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

Recall from part (a) that we have our linear system of the form $A\mathbf {x} =\mathbf {b}$ is: ${\begin{aligned}{\begin{bmatrix}{\frac {1}{3}}&{\frac {1}{3}}&{\frac {1}{3}}\\\\0&{\frac {1}{2}}&{\frac {1}{2}}\\\\{\frac {1}{2}}&{\frac {1}{2}}&0\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\end{bmatrix}}&={\begin{bmatrix}12\\10\\16\end{bmatrix}}\end{aligned}}$ Therefore, we form an augmented matrix by adding the output vector b as a new column in A. Therefore the augmented matrix is $\left[{\begin{array}{ccc|c}{\frac {1}{3}}&{\frac {1}{3}}&{\frac {1}{3}}&12\\&&&\\0&{\frac {1}{2}}&{\frac {1}{2}}&10\\&&&\\{\frac {1}{2}}&{\frac {1}{2}}&0&16\end{array}}\right]$