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# Linear Algebra

### Algebra Questions

See attached file for full problem description. 24. Complete the ordered pairs so that each is a solution for the given equation. 3x + 4y = 12 46. Science and medicine. Celsius temperature readings can be converted to Fahrenheit readings using the formula F = 9/5C + 32 . What is the Fahrenheit temperature that corresponds

### Solving Inequalities and Linear Systems of Equations

1. Find the solution to the following system of linear equations: 3x + 4y = -8 y = -5x - 7 2. My friend is driving 55 mph, West, on Interstate 90 and he started from his home which is 75 miles west of mine. I leave at the same time, traveling West at a speed of 73 mph. Write 2 linear equations (one for me and one fo

### Logic : Decidable Systems

Question Let (N, <) be the model with universe N and the "less than" relation. Show that Th(N, <) is decidable. Solution Reduce Th(AJ, <) to Th(fV, +), which we've already shown to be decidable. To do so, show how to convert a sentence 0, over the language of Th(A(, <), to a sentence 02 over the language of Th(AF, +) w

### Systems of Linear Equations

1. Solve the following systems of linear equations. Solve one using addition and one using substitution. State which method you are using for which system... a) 4x - 3y = 10 & y - 20 = -6x b) y = 3x + 6 & -6x + 2y = 12 2. You are buying coffee beans for your new business, "The Algebra Café." You

### Solving Systems of Linear Equations using Matrix Methods

Matrices have a number of interesting mathematical attributes, such as their dimensions, how they can be derived from linear systems, and the kinds of operations that can be performed on them. 1. Matrix methods can be used to solve linear programming problems. A linear programming problem is used to find an optimal solution,

### Evaluating Functions and Systems of Linear Equations and Inequalities

(See attached file for full problem description) 1. Graph f(x) = -2x + 2. 2. Rewrite the equation -x - 10y = 50 as a function of x. 3. Given f(x) = 5x2 - 3x + 1, find f(-2). A) -13 B) 15 C) -25 D) 27 4. Given f(x) = 4x - 5, find f(a - 1). A) 4a - 9 B) 4a - 6 C) a - 2 D) a - 9 5. Graph the inequality.

### Solving Systems of Differential Equations

Step by step process in solving the following question. Using matrix algebra techniques, find a general solution of the system: x'=3x-4y y'=4x-7y

### Linear Equations and Inequalities

1) Given one point on the graph (-3, -5) and the slope, m = -2, develop an equation in the slope-intercept format Lecture). 2) Given one point on the graph (- 2/3, 2/5) and the slope, m = - 3, develop an equation in the slope-intercept format(Lecture). 3) Given the following two points, (- 4, - 2) & (-1, - 5),develop an

### Solving Linear Systems of Equations : Gaussian Elimination and Cramer's Rule

Solve the following linear system using Gaussian elimination. Show work. 2y + z = 4 x+ y +z = 6 2x + y + z = 7 Solve the following linear system for x using Cramer's rule. Show work. 4x - y + z = -5 2x + 2y + 3z = 10 5x - 2y + 6z = 1

### Matrices

Matrix methods can be used to solve linear programming problems. A linear programming problem is used to find an optimal solution, subject to stated restraints. One typical application is to maximize profits. For example, a beauty parlor provides both highlighting and permanent wave services. It costs \$5 in materials and req

### Linear Equations

1. You are given the following system of linear equations: 3x - 2y + z = 2 -x + y = 3 -2y + 6 = -1 Provide a coefficient matrix corresponding to the system of linear equations. What is the inverse of this matrix? What is the transpose of this matrix? Find the determinant for this matrix. Calculate the following

### Solving Systems of Linear Equations by Graphing and Substitution / Addition Methods

Please see the attached file for the fully formatted problems.

### The Solution of a System of Equations

1. How do you interpret the solution of a system of equations by the corresponding graph? Please provide example. 2. What is the situation when two linear inequalities have no solution? 3. What are the conditions when the solution of a system of linear inequalities will be in the first quadrant?

### Explain Matrices

Matrices have a number of interesting mathematical attributes, such as their dimensions, how they can be derived from linear systems, and the kinds of operations that can be performed on them. Copy the questions to a Microsoft Word document and use an equation editor to enter the answers. Please answer the following question

### Solving the linear system using Gaussian elimination

Matrices are the most common and popular way to solve systems of equations. Provide an example of a matrix that can be solved using Gaussian elimination. Show specifically how row operations can be used to solve the matrix. State the solution. Substitute the solution back into the equation to verify the solution. O

### expressible as the union of conjugacy classes

Let G be a group, and let H be a subgroup of G. Prove that H is a normal subgroup if and only if H can be expressed as the union of conjugacy classes of G.

### General vector spaces - rank and nullity

10. Let A = a11 a12 a13 Show that A has rank 2 if and only if one or more of a21 a22 a23 the determinants a11 a12 a11 a13 a12 a13 a21 a22 a21 a23 a22 a23 are non zero. 14. Use the result in Exercise 10 to show that the set of points (x, y, z) in R3 for which the matrix

### Dependency Equation and Feasibility Analysis

4. The Volta Battery Company manufactures AA, A. C, and D batteries in each of four plants. The daily production (in 1000's) for the four plants is given the following table: [TABLE] (a) The vectors that represent the production at the various plants are not linearly independent. Show this and find a dependency equation. (b)

### Using Matrix to Solve Linear System of Equations

See attached file for full problem description. 1. Find the equation of the line shown: (see attachment). 2. A bank loaned \$15,000, some at an annual rate of 16% and some at an annual rate of 10%. If the income from these loans was \$1800, how much was loaned at 10%? 3. Write the augmented matrix of the system: (see attach

### Algebra

Please see the attached file for complete description 1. Solve the system by graphing. x + y = 4 -x + y = 2 2. Determine which two equations represent perpendicular lines. 3. Solve the system by graphing. 3x + y = 6 3x - y = 0 4. Graph the inequality y < -3 5. Solve the following system of linear inequalit

### General vector spaces - basis and dimension

10. Find the coordinate vector of p relative to the basis S = {p1, p2, p3}. (a) p = 4 - 3x + x2; p1 = 1, p2 = x, p3 = x2 (b) p = 2 - x + x2; p1 = 1 + x, p2 = 1 + x2, p3 = x + x2 22. Find the standard basis vectors that can be added to the set {v1, v2} to produce a basis for R4. v1 = (1, -4, 2, -3), v2 = (-3, 8, -4,

### Prove that there is no field of order 6.

Prove that there is no field of order 6. (You can not refer to the fact that all finite field are of prime-power order)

### General Vector Spaces - Linear Independence

2. Which of the following sets of vectors in R3 are linearly dependent? a) (4, -1, 2), (-4, 10, 2) b) (-3, 0, 4), (5, -1, 2), (1, 1, 3) c) (8, -1, 3), (4, 0, 1) d) (-2, 0, 1), (3, 2, 5), (6, -1, 1), (7, 0, -2) 4. Which of the following sets of vectors in P2 are linearly dependent? a) 2 - x + 4x2, 3 + 6x + 2x2, 2

### Systems of Linear Equations Word Problems

1. Leon drove 270 to the lodge in the same time as Pat drove 330 miles to the lodge. If Pat drove 10mph faster than Leon, how fast did each of them drive? 2. A company specializing in magazine sales over the telephone found that in 2500 phone calls, 360 resulted in sales and were made by a male caller, and 480 resulted in sal

### Using Matrix Inverses to Solve Systems of Linear Equations

A) Find the inverse of: |2 1 1| |0 1 2| |2 2 1| b) Use it to solve the following system of equations: 2x1 + x2 + x3 = 4 x2 + 2x3 = 0 2x1 + 2x2 + x3 = 1.

### Solving Systems of Linear Equations

Determine whether this system has a unique solution, no solution, or infinitely many solutions. If a solution exists, write it down. y = x x + z = 4

### Linear Equations:Predicted sales of a company

Suppose the sales of a company are given by S(x) = \$200x + \$20,000 where x is measured in years and x = 0 corresponds to the year 1996. Find the predicted sales in the year 2002 assuming the trend continues.

### Graphing and Solving Systems of Linear Equations

See attached file for full problem description. Graph y = -1. Solve the system by addition or substitution. 3x + 6y = 0 x = Find the slope of the line passing through the points (-4, 10) and (-6, 5). Solve: 6x + 4(3x - 2) = 6x + 2 Graph 3x + y = 3. Let y vary directly with x, with

### Linear Equation

(1) Given a line containing the points (1, 4), (2, 7), and (3, 10) determine the slope-intercept form of the equation and provide one additional point on this line, and graph the function.

### Index of subgroup and coset of subgroup

Explain what the index of a subgroup and a coset of a group are. Also, prove that if N is a subgroup of a group G such that [G: N] = 2, and if "a" and "b" are elements of G, then the product "ab" is an element of N if and only if either (1) both "a" and "b" are elements of N or (2) neither "a" nor "b" is an element of N.