Mathematical Analysis I, MATH 181 Final 1. Represent Each ✓ Solved

Mathematical Analysis I, MATH 181 Final 1. Represent each

1. Represent each pair of statements as a system of two equations. Verify the given values for x and y are a solution of the system of equations. (a) The sum of two numbers is twenty-five and twice the first number added to the second number totals thirty-two. Let x be the first number and y be the second number. x = 7 and y = 18. (b) An envelope of $10 and $20 bills contains eight bills and the money in the envelope is worth $110. Let x be the number of $10 bills and y be the number of $20 bills. x = 5 and y = 3.

2. Solve each equation by graphing. If the solution is not unique, identify the system as ‘inconsistent’ or ‘dependent’.

3. Solve each equation by the substitution method. If the solution is not unique, identify the system as ‘inconsistent’ or ‘dependent’.

4. Solve each equation by the elimination method. If the solution is not unique, identify the system as ‘inconsistent’ or ‘dependent’.

5. Find the augmented matrix representing the system of equations.

6. Find the system of equations represented by the augmented matrix. Add or subtract the following matrices. Multiply the following matrices.

7. Solve each system by row-reducing the corresponding augmented matrix. If the solution is not unique, identify the system as ‘inconsistent’ or ‘dependent’.

8. For each matrix A, find the inverse matrix, A-1, or identify A as a singular matrix.

Paper For Above Instructions

Mathematical analysis plays a crucial role in understanding complex systems and solving various mathematical problems. This paper will address the assignment requirements by representing each pair of statements as a system of equations, verifying the provided solutions, and employing various methods to solve the equations as requested.

1. Representation of Statements as Equations

(a) The first statement entails two numerical relationships: the sum of the numbers being twenty-five and the relationship between twice the first number and the second number totaling thirty-two. Setting up the system of equations:

Equation 1: x + y = 25

Equation 2: 2x + y = 32

With the solution x = 7 and y = 18, we can verify:

7 + 18 = 25 (True)

2(7) + 18 = 32 (True)

(b) The second statement deals with the number of $10 and $20 bills. Establishing the equations:

Equation 1: x + y = 8

Equation 2: 10x + 20y = 110

Substituting the solution x = 5 and y = 3, we verify:

5 + 3 = 8 (True)

10(5) + 20(3) = 110 (True)

2. Solving Equations by Graphing

Systems of equations can also be solved graphically which provides a visual understanding of the intersection points that represent solutions.

For Example:

Graphing the equations x + y = 25 and 2x + y = 32, we can plot the points and see where the lines intersect. This intersection gives us the solution to the equations.

3. Solving by Substitution Method

To solve using substitution, we can isolate one variable in one equation and substitute into the other:

From x + y = 25, we can express y:

y = 25 - x

Substituting into the second equation yields:

2x + (25 - x) = 32

Simplifying leads to the results to find x and y.

4. Solving by Elimination Method

The elimination method involves eliminating one of the variables by adding or subtracting the equations:

For example, adding the equations can help eliminate y in this example:

(x + y) + (2x + y) = 25 + 32

This simplification shall yield the values of x and, consequently, y.

5. Augmented Matrix Representation

The augmented matrix corresponding to the system of equations is essential in matrix operations:

For our first example:

| 1 1 | 25 |

| 2 1 | 32 |

6. System of Equations from Augmented Matrix

Transforming the augmented matrix back into a system of equations allows for interpretation. Operations such as row reduction enable the identification of solutions from the matrices.

7. Row-Reduction for Solution Identification

By performing row-reduction techniques, we can convert the augmented matrix into a form that reveals the solutions directly.

8. Inverse Matrix Identification

For each matrix defined in the problems, calculating the inverse can be crucial in determining unique solutions; singular matrices pose additional challenges.

Conclusion

This paper has demonstrated how to represent mathematical statements as systems of equations, verify solutions, and solve using various techniques including graphing, substitution, and elimination. Each method has its advantages depending on the situation and complexity of the equations involved.

References

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  • Burden, R. L., & Faires, J. D. (2016). Numerical Analysis. Cengage Learning.
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  • Spivak, M. (2018). Calculus. Publish or Perish.
  • Boyce, W. E., & DiPrima, R. C. (2010). Elementary Differential Equations and Boundary Value Problems. Wiley.

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