Science:Math Exam Resources/Courses/MATH152/April 2012/Question 02 (c)

MATH152 April 2012

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Question 02 (c)
Consider the following system of linear equations ${\begin{aligned}2x-y+2z&=a\\x+3y-6z&=b\\3x+2y-4z&=c,\end{aligned}}$ where a,b, and c $\in \mathbb {R}$. Give that x=y=z=1 is a particular solution to the system above, determine the values of a, b, and c. Then find all the solutions in this case and express them in parametric 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 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
Multiply the matrix with the vector of x, y, z.

Hint 2
Once we have a particular solution, how can we write the general solution using our work from part (b)?

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. If we multiply the matrix A to the vector [1,1,1] we get ${\begin{bmatrix}2&-1&2\\1&3&-6\\3&2&-4\end{bmatrix}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}={\begin{bmatrix}3\\-2\\1\end{bmatrix}}$ and therefore a=3, b=-2, and c=1. If we set those values permanently then we know that [1,1,1] is a particular solution to the problem. However, from part(b) we have that the solution to the homogeneous problem was $\mathbf {x} _{h}=t{\begin{bmatrix}0\\2\\1\end{bmatrix}}$ and therefore we can write that the general solution $\mathbf {x}$ is the sum of the particular and homogeneous solution, $\mathbf {x} ={\begin{bmatrix}1\\1\\1\end{bmatrix}}+t{\begin{bmatrix}0\\2\\1\end{bmatrix}}.$

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