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# Basic Algebra

### Algebra: Systems of Equations - 3 Word Problems

1. Joe has a collection of nickels and dimes that is worth \$5.65. If the number of dimes was doubled and the number of nickels was increased by 8, the value of the coins would be \$10.45. How many dimes does he have? 2. An express train and a local train leave GraysLake at 3 P.M. and head for Chicago 50 miles away. The express

### MATLAB : Least Squares - Solving Inexactly Specified Equations in an Approximation

The solution can ONLY be accepted in Matlab. The problem is in the attachment file. Least Square Planetary orbit [2]. The expression z = a + bxy + cy + dx + ey + f is known as a quadratic form. The set of points (x, y) where z = 0 is a conic section. It can be an ellipse, a parabola, or a hyperbola, depending on the si

### Exponentials, Simple and Compound Interest, Annuities, NPV and Amortization

1. Rational Functions Graph the following function when a=3 and b=2. Develop a generic expression (i.e., as a function of "a" and "b") to find the "x" and "y" intercepts for this function (see attachment) 2. Exponential Functions Once a new automobile enters the market, the manufacturers try to estimate their residual valu

### Divisbility

Prove that for all odd integers n, (1^n)+(2^n)+(3^n)...+(n^n) is a proper multiple of 1+2+3+...n

### Irrational roots

If a,b,c are odd integers, show that all real roots of ax^2+bx+c=0 are irrational numbers.

### Laws of Logarithms

a) log y = 1.2x -1 b) ln y = 1.2x - 1

### Algebra Topics: Radical Equation, Complete the Square etc.

See the attached file. 1. Solve the radical equation: {see attachment} 2. Find the equation of the line through the point (-2,-4) and perpendicular to the line {see attachment} 3. Complete the square, find the vertex, the axis of symmetry, all the intercepts, and graph the parabola: {see attachment} 4. ... Find the

### Correspondence of Upper Hemicontinuity

Please tell me whether or not the 2 correspondences are upper hemicontinuous and PLEASE (using the definition) justify why. 1)F:[2,3]->R^2, F(r)={(x,y):abs(x)+abs(y)<=r} 2)F:R^n{0}->R^n, F(x)=B(x;||x||), the closed ball centred at x with radius ||x||. Thanks Note: abs=absolute value is the complement ||x|| is d

### Logarithmic and trigonometric equations

Section on using the properties of algebraic, trigonometric, logarithmic and exponential functions to solve problems. 1) Solve the following equation: log5 X + log5(X-2) = log5 X 2) The half-life of a certain radioactive element is 100days. That means that after 100 days ½ of the radioactive substance wil

### A Combination of Algebra and Geometry : Grid Squares and Rectangles

Consider the following grid: 1) How many squares (of all sizes) are there in this grid? 2) How many rectangles (of all sizes) are there in this grid? 3) Can you generalize your results to an n by n grid? Can you generalize further? Please see attachment for grid.

### Classify the Polynomials

Classify the 15 given polynomials as monomials, binomials, trinomials, and polynomials. Use the format given below for categorizing the polynomials. [Note: Simplify wherever possible] (Please see attachment for polynomials)

### Laws of Exponents and Opposites of Poynomials

1. Using one of the laws of exponents, prove that any number raised to the power 0 is 1. 2. You are given the following polynomial: 2x7 - 4x3 + 3x. If x were replaced with its opposite in each of the terms of the given polynomial, will it result in the opposite of the polynomial? Explain why or why not and illustrate to supp

### How to use Monte Carlo Method

Please describe how to use the Monte Carlo method to estimate the attached expression.

### Finite and Infinite Summations

a) Write an expression in sigma notation as shown for suitable terms n1, n2, and ai, for each of these finite and infinite summations. b) Use the method of telescopic sums to evaluate these finite sums, using the indicated anti-difference sequences. c) Which of the following sequences are null? Explain your answer. See

### Sum of 2 nonnegative numbers is 20 Find the numbers

The sum of 2 nonnegative numbers is 20. Find the numbers if: a)if the product of one number and the square root of the other is to be as large as possible, and b)if one number plus the square root of the other is to be as large as possible.

### Find all solutions of each of the following congruences : (a) x2 + x +1 &#8801; 0 (mod 11) (b) x3 + x + 1 &#8801; 0 (mod 13) (c) x4 + x3 + 2 &#8801; 0 (mod 7)

Number Theory - Polynomial Congruences Find all solutions of each of the following congruences . (a) x2 + x +1 &#8801; 0 (mod 11)

### Binary Tree Induction

# Recall that a binary tree can be defined recursively as: * A Binary Tree is either empty * or A Binary Tree consists of a node with a left and right child both of which are Binary Trees. The degree of a node in a tree is equal to 0 if both children are empty, 1 if one of the children are empty, and 2 of both children ar

### Grammar Induction

Consider the grammar 1) <Number> -> <Digit><Number>|epsilon 2) <Digit> -> 0|1|2|3|4|5|6|7|8|9 Use induction to show that the number of strings in L(<Number>) of length n is equal to 10^n

### Aplication of Algebraic Expressions

3. Lisa is coloring her creation. The blue color used is 50% of the green color used, yellow is 5% of the blue color used, and brown is 7% of the amount of green used. She wants to find the total amount of color used by her. a. Find the total amount of color used by Lisa using an algebraic expression. b. Comment on the kind of

### Application of Algebraic Expressions

2. Mr. Pinto is a Web designer. He is trying to put 4 characters on a page. The space occupied by character 2 takes 5 units more than character 1, character 3 takes 7 units less than the character 2, and character 4 takes the same space as that taken by character 2. a. Find the total space occupied by all the characters in an a

### Creating algebraic expressions

1. Three prizes are to be distributed in a Creative Design Talent Search Contest. The value of the second prize is five-sixths the value of the first prize, and the value of the third prize is fourth-fifths that of the second prize. a. Express the total value of the three prizes as an algebraic expression. b. Comment on the ki

### Compound Inequality, Selling Price Range, and Retirement Pay

See the attached file. Solve each compound inequality and write the solution set using interval notation. 80) 0< 5-2x <=10 -6< 4-x<0 81) -3 < (3x -1)/ 5 < ½ 1/3 < (3-2x)/6 < 9/2 86) Selling Price Range: Renee wants to sell her car through a broker who charges a commission of 10% of the selling price. The book val

### Clearly identify the variables, coefficients, and constants

Translate the following into algebraic expressions. Clearly identify the variables, coefficients, and constants: a) One fourth of one half of the product of two numbers b) Two-thirds of the quotient of two numbers c) The product of 7 and twice n

### Variables

Lydia takes ten minutes more to complete an illustration than Tom. The total time taken by both of them is six hours. Form an algebraic equation to express this and identify the variables, coefficients, and constants of the algebraic expression. You are not required to solve the equation.

### Measurement of Angles

1. In a certain triangle the measure of one angle is double the measure of a second angle but is 10 degrees less than the measure of the third angle. [The sum of the measures of three interior angles of a triangle is always 180 degrees.] Form an algebraic equation to express the problem and identify the variables, coefficients,

### Let G, H be graphs such that G is a subgraph of H. Prove or disprove each of the following: (a) alpha(G) <= alpha(H) (b) alpha(G) >= alpha(H) (c) omega(G) <= omega(H) (d) omega(G) >= omega(H)

The stability number, alpha(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that no two of the vertices in S are connected by an edge of G. The clique number, omega(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that every pair of

### Syntactic categories

Write productions that will define the syntactic category <SimpleStat>, to replace the abstract terminal simpleStat in the figure below {see attachment}. You may assume the syntactic category <Expression> stands for C arithmetic expressions. Recall that a "simple statement" can be an assignment, function call, or jump, and that

### 25 Algebra Problems : Graphs, Word Problems, Equations, Inequality, Quadratic Equations

Please remember to show your methodology as well as the answer. How you solved the problems remember to follow logical rules AND PLEASE BE CAREFUL - CHECK YOUR ARITHMETIC WATCH OUT FOR THOSE QUESTIONS THAT NEED ± SOLUTIONS There are 25 questions. For questions 1 through 9, perform the indicated operations. If possib

### Proof about congruence modulo 43 (also expressible as equivalence modulo 43)

Let S = Z_43 (where the underscore, "_", indicates that what follows it, in this case 43, is a subscript). Let Q be a subset of S that contains ten non-zero numbers (i.e., that Q contains ten non-zero elements of S). Prove that Q contains four distinct numbers "a," "b," "c," "d" such that ab = cd in Z_43.

### Lowest Common Multiple Application Word Problem

Five children collect N pieces of Halloween candy and decide to split it evenly among them. When they try to divide it they have two pieces of candy left over. One of the children leaves, taking the 26 pieces of candy she collected with her. The remaining four children try to split the N-26 remaining pieces of candy and discover