MAT2705-02/05 15S Quiz 4 Print Name (Last, First) _________________________________|__Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use arrows and equal signs when appropriate. Always SIMPLIFY expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). Indicate where technology is used and what type (Maple, GC).1. 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
a) Write down the coefficient matrix A, the RHS matrix 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 and the augmented matrix C = <A | 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> for this linear system of equations.b) With technology (identify your choice!), reduce this matrix C step by step to its ReducedRowEchelonForm avoiding fractions (7 steps!), recording the intermediate matrices and row operations for each step (as in 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). You may combine the AddRow operations within a single pivot, reporting only the final matrix.
c) Write out the equations that correspond to the reduced matrix. Identify the leading variables and the free variables and solve. State your solution in the scalar form: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRjw= = ..., LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj4vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0ZARj4= = ..., etc .
d) Does it agree with the technology solution of the original system? If not, there must be a mistake somewhere.
2. a) Use technology to solve the following system (recall the instruction to identify your choice of technology, and perhaps you should rewrite these equations first lining up the variables):
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
b) This has a lowest (positive) integer solution representing the balance of the chemical reaction 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. Write out this lowest integer solution and check that it satisfies the final equation of this system.solution1.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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LinearAlgebra:-ReducedRowEchelonForm( );LinearAlgebra:-LinearSolve( );Student:-LinearAlgebra:-LinearSolveTutor( );so the variables are:
Free: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYjUSdgeF9fM2BGJy1JI21vR0YkNi1RIixGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2I1EnYHhfXzRgRidGOEYrRjg=
Leading: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EnYHhfXzFgRictSSNtb0dGJDYtUSIsRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiNRJ2B4X18yYEYnRi8tRiw2I1EhRidGMw==and the solution 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This is completely standard strategy involving no swaps, just a series of pivots down the main 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with the Student LA commands: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 technology solutionchemical reaction link: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 If we insist that we get 4 water molecules out, how many of the others do we need to have to make the reaction work out. This gives us a nonhomogeneous linear system.
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