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# Linear Algebra

### Solving Systems of Equations with Three (3) Variables (Unknowns)

2x + y + 2z = 3 x + 6y + 3z = 4 3x - 2y + z = 0

### Graph of the function is to be identified

Please see the attachment for the description of the problems.

### Single variable equations

1) Find the degree measure of the angel marked Z 2) High-Risk Funds Of the \$50,000 that Natasha pocketed on her last real estate deal, \$20,000 went to charity. She invested part of the remainder in Dreyfus New Leaders Fund with an annual yield of 16% and the rest in Templeton Growth Fund with an annual yield of 25%. If she m

### System of equations

4. The St. Marks community bbq served 250 dinners. a child's plate cost 3.50 and an adult's plate cost 7.00. A total of 1347.50 was collected. how many of each type of plate was served? 8. Deep thought granola is 25% nuts and dried fruit. Oat dream granola is 10% nuts and dried fruit. How much of deep thought and how much of

### Let T be a tree of order at least 4, and let e1, e2, e3 belong E(T^). Prove that T+e1 +e2+ e3 is planar.

Let T be a tree of order at least 4, and let e1, e2, e3 belong E(T^), T^ means Complement of T. Prove that T+e1 +e2+ e3 is planar. Please, draw the graph and explain it step by step.

### Diagonalizable Matrices, Image, Kernels and Direct Sums

Prove that if T &#1028; L(V) is diagonalizable then V = im(T) + ker(T) (+ = direct sum) (Hint: Use a basis of eigenvectors. The eigenvectors of the eigenvalue zero are a basis for the null space, and the remaining eigenvectors are a basis for the image) See attached file for full problem description. keywords: matrix

### Rings : Annihilators

(5) Let R be a ring with 1 and M a left R-module. If N is a submodule of M, the annihilator of N in R is defined to be: {r in R/rn=0 for all n in N} Prove that the annihilator of N in R is a two-sided ideal of R.

### Solving Systems of Linear Equations by Graphing

Solve the system of equations graphically. Then classify the system as consistent or inconsistent and the equation as dependent or independent. 6. 2y = 6 - x, 3x - 2y =6 14. y - x = 5, 2x - 2y = 10 16. y = 3 - x. 2x + 2y = 6

### Graphing and Solving Systems of Equations and Word Problems

Please see the attached file for the fully formatted problems.

### Sequences and Series: Using the index of a series as the domain and the value of the series as the range, is a series a function?

UNIT FIVE DB B: Using the index of a series as the domain and the value of the series as the range, is a series a function? Include the following in your answer: ? Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the arithmetic series? ? Which one of the basic functions (linear,

### Conjugacy Classes

Let K={k1,....km} be a conjugacy class in the finite group G. a) Prove that the element K=k1+k2+....km is the center of the group ring R[G] (check that g^-1Kg=K for all gin G) b) Let K1,....Kr be the conjugacy classes of G and for each Ki let Ki be the element of R[G] that is the sum of the members of Ki. Prove that an el

### Applications of Linear Equations

1. An art dealer sold two artworks for \$1520 thereby making a profit of 25% on the first work and 10% profit on the other, whereas if he had approached any exhibition he would have sold them together for \$1535 with a profit of 10% on the first and 25% on the other artwork. Find the actual cost of each artwork. 2. A professor

### Systems of Equations

Find a linear function f(x) = mx + b whose graph has the given slope and y-intercept. 6. Slope: 4/5 ; y-intercept: (0,28) Find an equation of the line having the given slope and containing the given point 14. m = 3, (-2, -2) Find an equation of the line containing the given pair of points. 28. (-4, -7) and (-2, -1

### Solving System of Linear Equations Characteristics

1. Find k if the following system of equations has a unique solution 2x + (k - 1)y = 6 3x + (2k + 1)y = 9 2. Find k if the following system of equations has infinite solutions kx + 3y = k - 3 12x + ky = k 3. Find k if the following system of equations has no solution 3x + 2y = 6 kx + (k - 1)y = 9

### Solve for F

See attached file for full problem description. When converting from Fahrenheit degrees to Celsius degrees, a well known formula is used: C= 5(F - 32) / 9

### Coordinate system

See attached file for full problem description. 1. Find the slope of the line passing through the points (-8, -3) and (-2, 2). A) B) C) D) 2. Give the coordinates of the point graphed below. A) (4, 0) B) (-4, 0) C) (0, 4) D) (0, -4) 3. Find the slope of the graphed line.

### Why every odd order subgroup has to be subgroup of cyclic odd

Please see the attached file first. I did the proof, but it's weak, since I can't find a way to argue why every odd order subgroup has to be a subgroup of the cyclic odd order subgroup K of index 2, and by the divisibility argument it still could be a subgroup of G and not to be contained entirely in K.

### Solving Systems of Linear Equations Word Problems

Jill, Karen, and Betsy studied a total of 93 hours last week. Jill's and Karen's study time totaled only one-half as much as Betsy's. If Jill studied 3 hours more than Karen, then how many hours did each one of the girls spend studying?

### Solve each system by the addition method

3x/2 - 2y/3 = 10 1/8x + 1/4y=5

### Matrices, Eigenvectors, Eigenvalues and Inverses

Determining the equation for a matrix and confirming the inverse, eigenvalues Details: I have think the answer to question (a) is theta^2 [(1-p)I + pJ]. But when I am multiplying the sigma and the sigma inverse I still have "stuff" at the tail end of the identity......being added. I know I should be left with the identity if t

### Prove that every regular tournament is strong.

Prove that every regular tournament is strong. Hints Here we need to first figure out something more about outdegrees and indegrees and orders of regular digraphs. Try to find a regular digraps with 3, 4, 5 ,6 vertices, and generalize. D is a regular tournament if there is k such that outdegree x = k and indegree x = n-k

### Solving Systems of Linear Equations Word Problems

Flying against the jetstream, a jet travels 1890km in 3 hours. Flying with the jetstream, the same jet travels 4650km in 5 hours. What is the speed of the jet in still air, and what is the speed of the jetstream?

### Systems of Equations Word Problems

Two rainstorms occurred in one week in a certain area. In the first rainstorm 35ml of rain fell per hour, and in the second rainstorm 20ml of rain fell per hour. Rain fell that week for a total of 45 hours for a total rainfall of 1200ml. How long was each of the two rainstorms?

### Graphing Linear Equations

Draw the line whose y-intercept is -7 and whose x-intercept is 4.

### Forest and subgraph

3.2 prove that a graph G is a forest if and only if every induced subgraph of G contains a vertex of degree at most 1. Please can you explain in here when the graph G is a forest and induced subgraph.

### Systems of equatinos

1. (Solve system by the substitution method. Determine whether the equations are independent, dependent or inconsistent.) 2x - y = 3 2y = 4x - 6 2. (Solve system by the substitution method. Determine whether the equations are independent, dependent or inconsistent.) y = 3(x - 4) 3x - y = 12 3. (Solve sy

### Solving Systems of Linear Equations: Example Problem

Solve each system by the addition method. Indicate whether each system is independent, inconsistent, or dependent. 1. -3x + y = 3 2x - 3y = 5 2. x + 3(y - 1) = 11 2(x - y) + 8y = 28 3. 1/3x - 1/6y = 1/3 1/6x + 1/4y = 0

### Solving Word Problems using Systems of Equations

Flying against the jetstream, a jet travels 2600km in 4 hours. Flying with the jetstream, the same jet travels 6060km in 6 hours. What is the speed of the jet in still air, and what is the speed of the jetstream?

### Systems of Equations Word Problems - Motorboat

A motorboat travels 485KM in 5 hours going upstream and 714 km in 6 hours going downstream. What is the speed of the boat in still water, and what is the speed of the current?

### Systems of Equations Word Problems

A child in an airport is able to cover 380 meters in 4 minutes running at a steady speed down a moving sidewalk in the direction of the sidewalk's motion. Running at the same speed in the direction opposite to the sidewalk's movement, the child is able to cover325 meters in 5 minutes. What is the child's running speed on a sti