Operations Research: Optimal Pricing Strategies

• Operations Research Analysts create pricing strategies in various sectors including airline, computer, finance, healthcare, manufacturing, & transportation.
• Large corporations optimize product prices for high profits & quick sales.
• Manufacturers face the challenge of balancing price to ensure profit and sales.

Finding the Optimal Price

• Analysts determine the best car price by solving simultaneous production to sales equations.
• Where 'p' is the price of a car and 'n' is the number sold:
• A linear production equation indicates 'n' rises as 'p' rises.
• A linear sales equation indicates 'n' falls as 'p' rises.
• The best price enables the number of manufactured cars to equal definite sales, solving for (n, p).

Chapter Contents

• The chapter covers algebraic expressions, simplification, expanding expressions, and linear equations.
• Linear equations focus on pronumerals on one side, brackets, and both sides expressions.
• Also inlcuded are steps on solving word problems, linear inequalities and using formulas.
• Using linear and quadratic equations to solve problems.

NSW Syllabus Key points

• Develop fluency in math by connecting concepts and applying techniques for coherent communication.
• Simplify algebraic fractions with indices, and expand/factorize algebraic expressions (MA5-ALG-P-01).
• Solve monic quadratic, linear inequalities, and cubic equations (MA5-EQU-P-01).
• Solve linear equations with more than 3 steps, monic/non-monic quadratics & simultaneous equations (MA5-EQU-P-02).

Importance of Algebra

• Algebra underpins mathematics and problem solving in theoretical and practical scenarios.
• Variables are used in algebra to represent unknown quantities.
• Dentists use algebraic formulas to determine the quantity of local anaesthetic required, the formula represents anaesthetic concentration and patient weight.

Key Algebraic Concepts

• Letters in algebra represent numbers, terms are pronumerals or variables.
• An expression is a combination of numbers & pronumerals linked by +, -, x, and ÷, including brackets.
• A term combines numbers & variables via multiplication/division, separated by + or -.
• Coefficients are numbers multiplying pronumerals: e.g., 3 in 3x, -2 in 5-2x.
• Constant terms are just the number.

Evaluating Expressions and Order of Operations

• Evaluate an expression via pronumeral substitution.
• To substitute -2 into x+6 gives -2+6 = 4.

BODMAS

• Follow an order of operations must be followed when evaluating expressions:
• Expressions in brackets first
• Powers
• Multiplication/division (left to right),
• Addition/subtraction (left to right).

Writing Algebraic Expressions

• The number of tickets needed for 3 boys and r girls can be represnted as 3 + r
• The cost of P pies at $3 dollars each can be represented as 3p.
• The number of grams of peanuts for one child when 300g of peanuts is shared equally among C children, can be represented as 300/C

Converting to Expressions

• Five less than x. This can be represented as x - 5
• The sum of a and b is divided by 4. This can be represented as (a + b) / 4
• The square of the sum of x and y. This can be represented as (x + y)²

Exercise 2A:

• Includes evaluating expressions by substituing values into equations.
• Includes real world scenarios like the cost of P pies at $6 dollars each is represented as 6P

Simplifying Algebraic Expressions

• Just as 2+2+2+2=42, so x + x + x + x =4 * x or 4x*, this expression can be simplified to 4x.
• A single term like 25x/ 10 is simplified using multiplication and division.

Multiplication and Division

• Multiplication (×) and division (÷) can be simplified
• 5 × a × b = 5ab and −7 × x ÷ y2 = -7x/y2
• Globe maker's algebraic expressions can be simplified to the revenue and profit.

Key Principles of Algebraic Terms

• Common parts of algebraic expressions can be cancelled during division.
• Like terms contain identical pronumeral factors, for example 5x & 7x.
• The pronumeral part of a term is usually written alphabetically.
• Like terms are combined through addition/subtraction to form a single term e.g 5ab + 8ab = 13ab
• Unlike terms differ in their pronumeral factors.

Multiplying Algebraic terms

• Multiply 3 * 2b by mulyiplying the coefficients to get 6b.
• Multiply -2a × 3ab by multiplying the coefficients and simplify to get -6a²b

Dividing Algebraic terms

• 6 *ab / 18 b Divide numerals and pronumerals separately with the view to cancel where possible ( a / 3 )
• 12a²b / 3ab . Write as fraction, cancel and follow a² = a * a (4a)

Collecting Like Terms

• 3 * x* + 4 - 2 x = 3 x - 2 x + 4 = x = x+4
• The sign on a term follows it ( 3x and -2x) to combine with coefficients: 3-2 = coefficients
• 8 ab² - 9 ab – ab² + 3 ba = 8 ab² – ab² – 9 ab + 3 ab = 7 ab² – 6 ab

Expanding Algebraic Expressions: The Basics

• The distributive law removes expanding brackets through multiplication.
• A factor outside brackets multiplies each term inside: a(b + c) = ab + ac.
• If a bracket is preceded by a negative, signs of the inner terms reverse upon expansion: −a(b + c) = −ab − ac.

Expanding and Simplifying Expressions

• Expressions can be expanded and simplified by use of the distributive law.
• 4(x + 3y) can be written as 4 * x + 4 * 3 * y* (x
• -2x (4x − 3) = −(2x) * 4x + (−2x) * (−3)= −8x² + 6x

More on expanding expressions

• Expanding Brackets: Multiply each term inside the brackets.
• 3(x - 4) is equivalent to 3x - 12
• Then combine any like terms (constant integers): for example 2 - 12 = -10.
• Exapnd each se to expression :−(3x + y) is equivalent to −1(3x + y) = − 3x - y.
• Collect like terms then simplify.

What is an equation?

• An equation includes = with both equivalent left/right sides.
• 5 = 10 ÷ 2 and 1/x are examples of an equation.
• Equations are solved when calculating the real variable amount.
• Equivalent equations can be formed when the same action has taken place on opposite sides of the =.
• An example of a linear equation is to write it in the form ax + b = c (1x).
• Complex linear equations require several process like using the order of operations.

Performing operations

• Add/subtract the side.
• Multiply/divide from both sides.
• Solving equations is equivalent to following reverse operations.
• Checking solutions = checking what the side values should equate to.
• Dividing cancels operations (a-7+7 or x/4*4).

Linear Equations with Fractions

• Eliminate fractions to solve these.
• Multiplying by 9 first can elimate fractions, then substract remaining side with 4 (to get the answer).

Linear Inequalities

• An inequality uses <, ≤, >, or ≥.
• E.g., 2 < 6, 5 ≥ − 1,3x + 1 ≤ 7 and 2x + 3 > 4 are examples of inequalities.
• They can represent an infinite range of values, such as 2x < 6 meaning x < 3.

Guidelines for Inequalities

• You can illusturate using a number line.

• (greater than) or < (less than) show use an open circle.

• Use a closed circle to illustrate ≥ (greater than or equal to) or ≤ (less than or equal to) numbers.
• Lines show where upper and lower values.
• "Solution sets" describe all numbers that meet an inequality.
• Reverse the sign when multiplying by a negative in the sides.
• Sign also reverses when transposing / swapping.

Solving Inequalities

• Solve similar way to linear equations.
• The solution set include find the number for a variable on an inequality's expression.
• Multiplying the sides can be simplified.
• Place on the left and reverse variable to inequality symbol.

What is a formula?

• A formula uses equations or rules that determine relationships between variables.
• Volume can increase / decrease.
• Use math to determine side quantities.
• For calculating various problems solve the list of variables such as the side lenght.
• Variables can be evaluated with certain equations and numbers.
• $A$, $F$ and $d$ make the "subjects" on what the formula relates.

The Subject of a Formula

• A variable listed as the subject usually goes on the left, such as $C = 2πr$ has $C$ as the subject of the formula.
• All values and sides can use equations to be transpose.