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# Basic Algebra

### 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,

### Finite Set of Embeddings

Let E be a finite extension of a field F. Show that any finite set of distinct embeddings of E into the algebraic closure of F is linearly independent over F.

### 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

### Prove that in any graph...

Prove that in any graph with two or more vertices, there must be two vertices of the same degree.

### Exponents and Logarithms Problems

Graph y = 2^-x for -2 <= x <= 2. Solve for x: 3^4x-5 = 27. Suppose \$1000 is invested at 6 percent annual interest, compounded monthly. How many dollars wil be in the account after: a) one year? b) five years? Write in exponential form (log_9)(27) = 3/2. Write in logarithmic form: 1296 = 6^4. At 5% interest comp

### ODEs with Constant Coefficients

Find the general solution of the ODEs attached (about 10 different problems involving differential equations with constant coefficients) Solve the IVPs attached.

### ODE's with Constant Coefficients

See the attached file. Y1: Find the general solution of the OEDs: y'''-3y"+3y'-y=0 y"=0 y"+y'-2y=0 5y"-10'=0 And about 8 other ODEs with IVP.

### Determine the Nonnegative Integers

Please see the attachment for problem related to nonnegative integer and my solution (needs to be edited and confirmed).

### 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.

### Odd Primes, Inverses and Wilson's Theorem

Assume p is an odd prime ... Please see the attached file for the fully formatted problems.

### 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

### Roots of a quadratic equation,.,

Explain why 2 and 3/2 cannot be the roots of 2y2 - 7y + 3 = 0

### Quadratic Equations: X/Y-Intercepts, Square Root Property and Interval Notation

1. For the quadratic equation y = cx² + dx + e (where c, d and e are constants), what is the x-coordinate of the vertex? 2. For the same quadratic equation as above, what is the y-intercept? 3. Solve by the square root property: 4x² = 100 4. Solve by completing the square: x² - 10x + 4 = 0 5. Solve by usin

### 15 Algebra Problems : Find Equation for Line; Describe Graph and Slope

Please see the attached file for the fully formatted problems. Q1. Solve {see attachment} Q2. Solve 9 - 7+15/5 - (6-3)4 Q3. Solve {see attachment} Q4. Find the equation for the line with slope 3/5 and passing through the point (4, 2). Q5. Solve |2x-5| >= 14 Q6. A store issued coupons worth 20% off from any pu

### Solve for Y in a Function

Hello, Stumped on this one. Need to solve for y. 10-6xy = 4y + 8

### 10 Algebra Questions

1. The width of a rectangle is 8 feet less than the length. If the area is 20 square feet, find the length and the width. 2. Solve the equation: x(x - 4) = 12 3. A boat travels 30 miles upstream against the current in the same amount of time it takes to travel 42 miles downstream with the current. If the rate

### Perfect square/cube/fifth power

INSTRUCTIONS TO OTA 1. typeset solutions 2. send solution as attachment 3. Use words to explain solutions. DO NOT RELY ONLY ON ALGEBRAIC MANIPULATIONS/ OR SYMBOLS. Find the smallest positive integer N such that N/2 is a perfect square, N/3 is a perfect cube and N/5 is a perfect fifth power.

### Perfect Integers

An integer 'n' is called 'perfect' if it equals the sum of all its divisors 'd' ... {see attachment for complete definition and example} Let 'a' be a positive integer. Prove ... {see attachment}

### Lowest Common Multiple (Prime Factorizations)

Let a and b be integers. A common multiple of a and b is an integer n for which a|n and b|n. We call an integer m the least common multiple of n provided (1) m is positive, (2) m is a common multiple of a and b, and (3) if n is any other positive common multiple of a and b, then n [greater than or equal to] m. The notation fo

### Partial Induction Proof of Cauchy's Integral Formula

See attached file...it is a full induction proof of Cauchy Integral Formula, with the base case step missing. All I have to do is show that it holds for "n=1", using the rest of the proof as an example...however i am having trouble showing it.

### Rearranging Equation Calculations

I has been too long since I worked with detailed equations. I need to find out what (a) would be in the attached formula.

### Proof by Induction

Prove by induction that (1 + 1/2)^n >= 1 + n/2 for all n in N

### Solve an inequality and express using interval notation.

Solve an inequality. Write a solution set using interval notation. X^2 - 3 X - 5 >/= 0 "( X^2 - 3 X - 5) is greater than or equal to 0" PS: Please use those symbols, Multiplication * , fractions /, Exponents ^, Roots sqrt( )or &#8730;( ), - &#8730;( ), Infinities - ∞, ∞, minus - or plus + For example, (

### Wireless Communication Problem Set

1. When the FCC conducted the original A and B block PCS auction, what were the companies that won the license for Cleveland Ohio? What is the MTA number for Cleveland? How many pops were covered by the license? 2. Who was the winner original high bidder(s) for the D-Block auction for Houston BTA? What was the BTA number? Wh

### 21 Algebra Problems : Factoring, GCF and Word Problems

SHOW ALL STEPS WITH SOLUTION. FACTOR OUT THE GCF IN EACH EXPRESSION. 1. 12X^4T + 30X^3T - 24X^2T^2 2. 15X^2Y^2 - 9XY^2 + 6X^2Y FACTOR EACH POLYNOMIAL 1. 3X^2 + 6X + 3 2. X^3 + X^2 - X - 1 3. 3A - 3B - XA + XB FACTOR OUT EACH POLYNOMIAL AND FACTOR OUT THE GCF 1. H^2 - 9HS + 9S^2 2

### Finance : McDonalds Stock and Trend Lines

Please see the attached file for the fully formatted problems. 1. Pick your favorite financial site on the web 2. Go there and get the stock price for McDonalds for the first trading day in March and April of this year 3. Develop a trend line slope (change in dollars per month) 4. Check the stock price for the first trad

### Lagrangian multiplier

Use Lagrangian multipliers to find the points on the ellipse with the equation x^2+xy+y^2=3 that are closest to and farthest from the origin