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Questions and Answers
What is the primary role of Operations Research Analysts in the context of pricing strategies?
What is the primary role of Operations Research Analysts in the context of pricing strategies?
- To minimize production costs for manufacturers.
- To set prices according to the manufacturer's preference.
- To ensure products are priced low enough for quick sales.
- To develop pricing strategies that maximize profits and sales. (correct)
In supply and demand equations, an increase in the price of a car always leads to a decrease in both the number of cars manufactured and the number sold.
In supply and demand equations, an increase in the price of a car always leads to a decrease in both the number of cars manufactured and the number sold.
False (B)
In the context of simultaneous equations for supply and demand, what condition indicates the best price point for achieving the highest profit?
In the context of simultaneous equations for supply and demand, what condition indicates the best price point for achieving the highest profit?
The number of manufactured cars equals the number of definite sales.
According to the chapter contents, mathematicians that find a selling price suitable for both the manufacturers and customers are called ______ Research analysts.
According to the chapter contents, mathematicians that find a selling price suitable for both the manufacturers and customers are called ______ Research analysts.
Match the following algebraic concepts with their definitions:
Match the following algebraic concepts with their definitions:
Which of the following expressions represents 'the sum of a and b is divided by 4'?
Which of the following expressions represents 'the sum of a and b is divided by 4'?
According to the order of operations, exponents should always be evaluated before any operations within brackets are performed.
According to the order of operations, exponents should always be evaluated before any operations within brackets are performed.
If $a = 3$, $b = -1$, and $c = -2$, what is the value of the expression $3(a - b) + 2c$?
If $a = 3$, $b = -1$, and $c = -2$, what is the value of the expression $3(a - b) + 2c$?
The expression $5 \times a \times b$ is written as ______ in simplified algebraic terms.
The expression $5 \times a \times b$ is written as ______ in simplified algebraic terms.
Match each expression with its simplified form:
Match each expression with its simplified form:
What is the result of $5x \times 7n$?
What is the result of $5x \times 7n$?
The expression $6ab \div 18b$ simplifies to $3a$
The expression $6ab \div 18b$ simplifies to $3a$
Simplify the expression $6a + 2 - a$
Simplify the expression $6a + 2 - a$
Expanding the expression $-2(x-5)$ results in ______.
Expanding the expression $-2(x-5)$ results in ______.
Match the following expansions with their original expressions:
Match the following expansions with their original expressions:
If $3x + 1 = 10$, which operation should be done first to find solution to x?
If $3x + 1 = 10$, which operation should be done first to find solution to x?
If $5 - 2x = 12$, then $x$ equals $\frac{7}{2}$.
If $5 - 2x = 12$, then $x$ equals $\frac{7}{2}$.
Solve for x: $\frac{x}{4} - 3 = 7$
Solve for x: $\frac{x}{4} - 3 = 7$
In solving linear equations with fractions, the first step often involves ______ sides of the equation by a common denominator to eliminate the fractions.
In solving linear equations with fractions, the first step often involves ______ sides of the equation by a common denominator to eliminate the fractions.
Match the equation with its solution for (x):
Match the equation with its solution for (x):
Which of the following is equivalent to writing $\frac{6 + b}{2} = 3$?
Which of the following is equivalent to writing $\frac{6 + b}{2} = 3$?
Given the equation $0 = \frac{6+b}{2} - 3$, then it makes sense to consider $b=-12$
Given the equation $0 = \frac{6+b}{2} - 3$, then it makes sense to consider $b=-12$
If it is known that (a + 8 = 4x ), and (j = 4 + x), combine the equations given knowing (x=9).
If it is known that (a + 8 = 4x ), and (j = 4 + x), combine the equations given knowing (x=9).
When solving the equation $3(2x -1) = 4$, after applying the distributive property, the next step is to add 3 to both sides, resulting in $6x = ______$.
When solving the equation $3(2x -1) = 4$, after applying the distributive property, the next step is to add 3 to both sides, resulting in $6x = ______$.
Match the equation from Exercise 2E to its next step in reaching a solved or simplified solution:
Match the equation from Exercise 2E to its next step in reaching a solved or simplified solution:
With the equation of the formula (A=953)), which of the following correctly describes whether that makes "finding c" equal to zero or infinity?
With the equation of the formula (A=953)), which of the following correctly describes whether that makes "finding c" equal to zero or infinity?
Linear equations always follow a similar pattern such that when a graph extends a linear line beyond the viewable window, they will always at a point, have a 'known or easily calculable' (such as through simple ratio calculations) 'X and Y intercept'.
Linear equations always follow a similar pattern such that when a graph extends a linear line beyond the viewable window, they will always at a point, have a 'known or easily calculable' (such as through simple ratio calculations) 'X and Y intercept'.
Using the equation to show that ((i) 31 - k){+ (5)} (. What was B.
Using the equation to show that ((i) 31 - k){+ (5)} (. What was B.
Before equations get too difficult. Often the primary key that determines solvability relies on ______, in many mathematical and physical situations.
Before equations get too difficult. Often the primary key that determines solvability relies on ______, in many mathematical and physical situations.
Combine the formula provided with the step next needed for it to compute, giving correct values:
Combine the formula provided with the step next needed for it to compute, giving correct values:
What concept is fundamental to both algebraic equations and real-world problem-solving scenarios?
What concept is fundamental to both algebraic equations and real-world problem-solving scenarios?
In mathematical word problems, success depends solely on identifying the type of equation needed and less so on accurately defining your variables before computing.
In mathematical word problems, success depends solely on identifying the type of equation needed and less so on accurately defining your variables before computing.
In an inequality, what action must be taken when multiplying or dividing both sides by a negative number?
In an inequality, what action must be taken when multiplying or dividing both sides by a negative number?
When illustrating an inequality on a number line, an open circle is used to indicate that a number is ______ included in the solution set.
When illustrating an inequality on a number line, an open circle is used to indicate that a number is ______ included in the solution set.
Match for the correct next step, from initial configuration:
Match for the correct next step, from initial configuration:
The formula $C = 2Ï€r$ gives:
The formula $C = 2Ï€r$ gives:
Transposing is a type of mathematical method for changing variables, and is fundamentally separate from concepts from physics which may be transposed but often have deeper implications about their nature.
Transposing is a type of mathematical method for changing variables, and is fundamentally separate from concepts from physics which may be transposed but often have deeper implications about their nature.
If a shape's area was known only in the formula ((b = ((423 - \pia) /a)) ^ (1/1)), what formula variable would need to be known to accurately to transposed into the variable value of a?
If a shape's area was known only in the formula ((b = ((423 - \pia) /a)) ^ (1/1)), what formula variable would need to be known to accurately to transposed into the variable value of a?
To transpose (h = 10 (m+2)) to make (v) the subject.
To transpose (h = 10 (m+2)) to make (v) the subject.
Match the steps for simultaneous equations with what step comes next:
Match the steps for simultaneous equations with what step comes next:
What algebraic method can solve simultaneous equations in most circumstances?
What algebraic method can solve simultaneous equations in most circumstances?
Once (x) is calculated across simultaneous equations, it helps to find a corresponding '(y)' using just the first equation.
Once (x) is calculated across simultaneous equations, it helps to find a corresponding '(y)' using just the first equation.
Following substitution, the following equations for (x) are presented: (15+X=30, J{d}=A{d] 2x), what was '(y)'s corresponding value if A[d]=5)?
Following substitution, the following equations for (x) are presented: (15+X=30, J{d}=A{d] 2x), what was '(y)'s corresponding value if A[d]=5)?
Both simultaneous and ______ mathematical concepts are needed by Engineers in practical, many physical situations.
Both simultaneous and ______ mathematical concepts are needed by Engineers in practical, many physical situations.
Match in how to deal with each of the following variable setups, knowing it has to get both the X and Y values:
Match in how to deal with each of the following variable setups, knowing it has to get both the X and Y values:
Flashcards
What are pronumerals or variables?
What are pronumerals or variables?
Letters that represent numbers in algebra.
What is an expression?
What is an expression?
Numbers and pronumerals linked with +, -, ×, or ÷.
What is a term?
What is a term?
Numbers and variables joined by multiplication/division only.
What are coefficients?
What are coefficients?
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What are constant terms?
What are constant terms?
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What does it mean to evaluate expressions?
What does it mean to evaluate expressions?
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What is the order of operations?
What is the order of operations?
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What are like terms?
What are like terms?
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How are equivalent equations created?
How are equivalent equations created?
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What is back tracking an equation?
What is back tracking an equation?
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What is a solution set of numbers?
What is a solution set of numbers?
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What is a formula?
What is a formula?
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What is the subject of a formula?
What is the subject of a formula?
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What does it mean to transport an equation?
What does it mean to transport an equation?
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What are linear simultaneous equations?
What are linear simultaneous equations?
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What is the substitution method?
What is the substitution method?
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What is the elimination method?
What is the elimination method?
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What are quadratic equations?
What are quadratic equations?
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Quadratic equations, is it possible?
Quadratic equations, is it possible?
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Study Notes
Expressions, equations and inequalities
- This chapter explores expressions, equations, and inequalities central to mathematics for problem-solving.
- Mathematicians specializing in Operations Research develop pricing strategies across various industries.
- Examples of industries; airline, computer services, financial engineering, healthcare, manufacturing, mining, transportation and the military.
- Global corporations seek high profits and quick sales through strategic product pricing.
- A car's price example shows the need to balance profit and sales.
Finding the optimal price
- Equations analyzing supply (production) and demand (sales) find the best car price.
- p equals the price of one car, and n equals the number of cars sold.
- A linear equation for supply indicates n increases as p increases.
- A linear equation for demand indicates n decreases as p increases.
- Solving supply and demand equations finds (n, p) that satisfy both.
- Highest profit point equates the number of cars made to the number sold.
Chapter Content Overview
- This chapter covers algebraic expressions, simplification, expanding, linear and quadratic equations, inequalities, formulas, and simultaneous equations.
- Topics to be explored; algebraic expressions, simplifying, expanding algebraic expressions, linear equations (with or without brackets), word problems, inequalities, formulas, and solving simultaneous equations.
NSW Syllabus Outcomes
- Chapter aims to develop math understanding and fluency, choose/apply techniques, and clear coherent communication.
- Students will simplify algebraic fractions, expand and factorize.
- They will solve quadratic, linear inequalities and cubic equations.
- Students will solve multi-step linear as well as monic and non-monic quadratic and simultaneous equations.
Algebraic Expressions (Consolidating)
- Algebraic expressions are reviewed, including terminology.
- Focus is on identifying terms, coefficients, and constants, and substituting values.
- This reviews Stage 4 concepts for Stage 5 and 6, revised for Year 10 and expected as prior knowledge for Stage 6 Advanced
- Algebra solves theoretical/practical problems through math representation of variables.
- Pronumerals or variables represent unknown quantities.
- Algebraic expressions can be converted from worded scenarios.
- Numbers can be substituted into alegbraic expression
Key concepts of algebraic expressions
- Letters represent numbers known as pronumerals or variables.
- Expressions combine numbers and pronumerals with +, -, ×, ÷, and brackets.
- Terms combine numbers and variables using multiplication/division separated by +/–.
- Coefficients multiply pronumerals.
- Constant terms include only a number with its sign.
- Evaluating involve substituting a number for a pronumeral.
- Operations order matters; Brackets, Powers, Multiplication/Division, Addition/Subtraction (BODMAS).
Simplifying algebraic expressions
- Like terms combined under addition, subtraction in simplified algebraic expressions
- Multiplication/division conventions are also reviewed
- The highest common factor can divide terms
- Past, present, future learning consolidates Stage 4 concepts, for Stages 5 and 6 revised very briefly in Chapter 1 of Year 10 book.
- Assumed expertise required for Stage 6 Advanced 2+2+2+2=4x2, or just as x+x+x+x=4 × x or 4x
Multiplication and division
- The symbols for multiplication (×) and division (÷) are usually not shown in simplified algebraic terms.
- Common factors are cancelled when dividing expressions
- The pronumeral part of a term is commonly written in alphabetical order
- Like terms combine in to forma single term
- Unline terms do not have the the same pronumeral factors
Expanding algebraic expressions
- Expand and remove brackets using the distributive law
- A term outside brackets multiplies each term inside
- The sign of each term inside the bracket will change when expanded if the number infront of teh bracket is negative
Building understanding
- Each term is multiplied inside the brackets
- Then like terms are combined
Linear Equations with Pronumerals on One Side
- Equations contain an equals sign, a left-hand side, and a right-hand side.
- Linear equations are written as ax + b = c, where x has a power of 1.
- Determine variables to solve an equation
Creating equivalent equations and back tracking
- Equivalent equations adding/subtracting both sides by the same number.
- Solve by inverse operations, known as backtracking.
- Substitute in original equation to check the solution.
Solving simple linear equations involving fractions
- The same funadamentals apply for linear equations involving fractions
- Remember order of operations
Steps in the correct order
- Equations can be solved by expanding brackets first.
- Collect like terms by adding or subtracting to one side.
- Brackets must be solved by following the order to maintain balance.
Working backwards
- The distributive law must be known
- Understand that all terms are pronumerals and apply the rules
- Collecting like terms is also important with this method
Solving problems
- Let x hours correspond to the number of hours of television watched by Rick.
- What is the expression for the number of hours of television watched by Kate and by Sue.
- An equation needs writing to represent the information previously expressed.
- Next the equation needs solving.
- Now answer the questions in the problems
Key ideas when solving problems
- Problem needs to be read and known what the question is asking for.
- The variable be the thing we are trying to find out
- You need an equation that shows the facts the question presents
- Check to see if the soylition is correct
Key ideas when solving inequality questions
- An if < is replaced with >, If multiplying or divinding both sides of an inequality with a negative number
General equations
- Remembering to write all the information down
- Draw a picture if nesscary to help with visualising what is presented
- Remember to define x with x = ? so that you don't have to repeat what it is every step
Key points and definitions
- A formula (or rule) is an equation that relates two or more variables. You can find the value of one of the variables if you are given the value of all others. Some common formulas contain squares, square roots, cubes and cube roots. The following are some examples of formulas.
- A, F and d are said to be the subjects of the formulas given above.
- The subject of a formula is a variable that usually sits on its own on the left-hand side.
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