Price for Highest Profit

• Mathematicians called Operations Research Analysts develop pricing strategies for various industries

Finding the Price for Highest Profit

• Large corporations want to price products for high profits and quick sales
• The manufacturer sets the price for an electric car
• Too low a selling price results in no profit
• A high price produces more profit because more cars are manufactured
• A high price only works as long as all cars are quickly sold
• To find the best price to satisfy manufacturers and customers, analysts solve equations for Supply (production) and Demand (sales)
• Best car selling price for highest profit occurs when manufactured cars equal sales
• p = price of one car
• n = number sold
• Linear equation for Supply = Supply shows that as p increases, n increases
• Linear equation for Demand = Demand shows that as p increases, n decreases

Algebraic Expressions

• Algebra solves theoretical and practical problems within mathematics
• Algebra involves unknown or varying quantities
• Pronumerals (or variables) represent unknown quantities

Key Ideas

• Letters represent numbers
• These letters are pronumerals, or variables
• An expression equals a combination of numbers and pronumerals connected by +, -, ×, ÷
• Examples: 5x² + 4y - 1 and 3(x + 2) - 5/x
• A term equals a combination of numbers and variables connected with only multiplication and division
• Terms are separated by + and -
• Example: 5x + 7y consists of a two-term expression
• Coefficients are numbers multiplied by pronumerals
• Examples: 3 in 3x, -2 in 5 – 2x, and ½ in ½x are coefficients
• Constant terms consist of only a number
• Example: -2 in x² + 4x − 2
• Inclusion of a sign is a must
• Expressions can be evaluated by substituting a number for a pronumeral
• Example: if a = −2 then a + 6 = −2 + 6 = 4
• Order of operations must be followed when evaluating expressions:
  • Brackets
  • Powers
  • Multiplication and division
  • Addition and subtraction

Simplifying Algebraic Terms

• The symbols × and ÷ generally aren't shown in simplified algebraic terms
• Example: 5 × a × b = 5ab, −7 × x × x ÷ y² = -7x²/y²
• When dividing algebraic expressions common factors can be cancelled

Like Terms

• Like terms have the same pronumeral factors
• Example: 5x and 7x; 3a²b and −2a²b
• ab and ba are like terms because a × b = b × a
• Alphabetical order is preferred for the pronumeral part of a term
• Like terms can be collected via adding and subtracting to form a single term
• Examples: 5ab + 8ab = 13ab and 4x²y - 2xy²- 2x²y = 2x²y
• Unlike terms do not have the same pronumeral factors
• Example: 5x, x², xy, and 4xyz/5 are unlike terms

Distributive Law to Expand & Remove Brackets

• A term directly outside brackets is multiplied by terms inside the brackets
• a(b + c) = ab + ac or a(b - c) = ab – ac
• -a(b + c) = −ab – ac or -a(b - c) = −ab + ac
• If a negative number sits in front of the bracket, the sign of each term inside bracket turns opposite when expanded
• Example: -2(x − 3) = −2x + 6 since -2 × x = -2x and -2 × (-3) = 6.

Solving Linear Equations

• An equation contains an equals sign, left-hand side, and right-hand side
• Example: 5 = 10 ÷ 2, 3x = 9, x² + 1 = 10 and 1/x = (5x - 2)/(x + 1) are equations X
• Linear equations can exist as ax + b = c where the power of x equals 1
• Examples: 4x − 1 = 6 and 3 = 2(x + 1) are all linear equations
• Finding the variable's value solves equations
• Inspection works for very simple linear equations -Example: in 3x = 15, x = 5 since 3 × 5 = 15
• Steps creating equivalent equations solve more complex linear equations

Key Ideas

• Equivalent equations are created by:
  • Adding or subtracting the same number on either side
  • Multiplying or dividing both equation sides by the same non-zero number
• Solve a linear equation by creating equivalent equations using inverse operations
• Backtracking works too
• Substitute solution into original equation and ensuring both sides are equal checks solution

Solving Equations with Brackets

• Equations with brackets can be solved by first expanding
• i.e. 3(x + 1) = 2 becomes 3x + 3 = 2 Adding or subtracting terms to one side collects pronumerals in an equation with pronumerals across both sides
• i.e. 3x + 4 = 2x – 3 becomes x + 4 = -3 by subtracting 2x from both sides

Solving Word Problems with Algebra

• 1. Read the problem and pinpoint the goal is to ask for.
• 2. Define a variable and its meaning
• 3. Write an equation using the variable showing the relationship between known facts.
• 4. Solve equations.
• 5. Answer the question in words.
• 6. Ensure solutions make sense.

Linear Inequalities

• An inequality utilizes symbol <, <, >, or >

• Examples: 2<6, 5>-1, 3x+17 and 2x + >

• Inequalities can be illustrated using a number line

• Open Circles indicate >/ greatest than or or < less than, where line is not included

• i.e o------->x

• Closed Circles indicate >/ great or equal to or </ less or equal to symbols, where line is included

• i.e -●----->X

• A set of numbers can have a low or upper bound, eg -2<3

• Linear inequalities are solved the same way as equations.

• Set all numbers that satisfy inequality are called 'Solution set'

Transposing Formulas

• Similar steps to solving equations, make a variable the subject by transporting it