Mathematical Analysis Imath 181 Final 1: Represent Each Pair

Mathematical Analysis Imath 181final1 Represent Each Pair Of Statem

Mathematical Analysis I, MATH 181 Final. Represent each pair of statements as a system of two equations. Verify the given values for x and y are solutions of the system. Solve each system by graphing, substitution, or elimination methods, and find the inverse of matrices where applicable.

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Mathematical analysis plays a crucial role in understanding and modeling various real-world phenomena. In this comprehensive examination, we focus on translating verbal statements into algebraic systems, solving these systems through different methods, and performing matrix operations including finding inverses. This approach not only solidifies understanding of fundamental algebraic techniques but also demonstrates their applicability across disciplines.

Part 1: Translating Statements into Systems and Verification

Initially, we convert verbal statements into formal systems of equations. For example, considering the first problem: "The sum of two numbers is twenty-five, and twice the first number added to the second number totals thirty-two," with x and y representing the first and second numbers respectively, yields the system:

  • x + y = 25
  • 2x + y = 32

Verification involves substituting the given solution, x=7 and y=18, into both equations to confirm their validity. Doing so shows that 7 + 18 = 25 and 2(7) + 18 = 32, confirming these values are solutions.

Similarly, another statement involves counting bills: "An envelope contains eight bills worth $10 and $20 each, totaling $110." Let x be the number of $10 bills and y the number of $20 bills, leading to the equations:

  • x + y = 8
  • 10x + 20y = 110

Substituting x=5 and y=3 confirms that 5 + 3 = 8, and 10(5) + 20(3) = 50 + 60 = 110, verifying the solutions.

Part 2: Graphical Solutions and System Nature

Graphing equations offers visual insight into solutions. For example, the system:

  • y = x + 2
  • y = -x + 4

Graphically, these lines intersect at a single point, indicating a unique solution. Systems with no intersection are inconsistent, implying no solutions, while overlapping lines denote dependent systems with infinitely many solutions.

Such analysis extends to other systems, emphasizing the importance of graphing in visualizing solution sets in algebraic systems.

Part 3: Solving Systems via Substitution

The substitution method involves solving one equation for one variable and substituting into the other. For example, given the system:

  • x + y = 4
  • 2x - y = 1

Solving the first for x yields x = 4 - y. Substituting into the second: 2(4 - y) - y = 1 simplifies to 8 - 2y - y = 1, or 8 - 3y = 1. Solving for y: y = (8 - 1)/3 = 7/3. Using x = 4 - y results in x = 4 - 7/3 = 12/3 - 7/3 = 5/3.

This method efficiently produces solution points, and constants thus obtained reveal the nature of the system—whether unique, inconsistent, or dependent.

Part 4: Eliminating Variables

The elimination method involves adding or subtracting equations to eliminate a variable. For instance:

  • x + y = 3
  • x - y = 1

Adding these gives 2x = 4, so x = 2. Substituting back into the first: 2 + y = 3, so y = 1. Such techniques are valuable for systems where substitution is cumbersome.

Part 5 and 6: Matrix Representation and Operations

Matrix representations codify systems efficiently. The augmented matrix for the system x + y = 4 and 2x - y = 1 is:

[1  1 |  4]


[2 -1 |  1]

Row operations reduce matrices to row echelon form to solve systems systematically. For example, subtract 2 times the first row from the second to eliminate x from the second equation:

[1  1 | 4]


[0 -3 | -7]

Back-substitution yields y = 7/3, and x = 4 - y = 5/3, confirming earlier solutions.

Adding matrices involves element-wise addition, while multiplication relies on dot products of rows and columns. Matrix inversion, when it exists, involves finding a matrix A^{-1} such that AA^{-1} = I, where I is the identity matrix. This process requires calculus of determinants, adjugates, and minors, critical in linear algebra applications.

Conclusion

Mastering the translation of verbal statements into algebraic systems, solving these via graphical, substitution, and elimination methods, and manipulating matrices are foundational skills with broad applications. These techniques enable rigorous analysis of mathematical models across sciences and engineering, emphasizing precision and analytical capability in problem-solving. The understanding and application of these concepts remain vital for advanced mathematical pursuits and real-world problem-solving contexts.

References

  • Anton, H., Bivens, I., & Davis, S. (2013). Calculus: Early Transcendentals (10th ed.). John Wiley & Sons.
  • Lay, D. C. (2012). Linear Algebra and Its Applications (4th ed.). Pearson.
  • Stewart, J. (2015). Calculus: Concepts and Contexts (4th ed.). Brooks Cole.
  • Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press.
  • Lay, D., et al. (2015). Linear Algebra and Its Applications (5th ed.). Pearson.
  • Kantorovich, L. V., & Akilov, G. P. (1982). Functional Analysis in Normed Spaces. Pergamon.
  • Winston, W. L. (2004). Operations Research: Applications and Algorithms. Dover Publications.
  • Haber, R. (2010). Matrix Analysis and Applied Linear Algebra. Springer.
  • Poole, D. C. (2015). Linear Algebra: A Modern Introduction (4th ed.). Cengage Learning.
  • Friedberg, S. H., Insel, A. J., & Spence, L. E. (2014). Linear Algebra (4th ed.). Pearson.

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