Deriving Equations
Deriving equations involves the process of systematically obtaining mathematical expressions based on given conditions, principles, or relationships. This often entails using logical reasoning, algebraic manipulation, and mathematical operations to arrive at a specific formula or relationship. The derived equations are essential for solving problems, making predictions, and understanding various phenomena in mathematics and its applications.
3 Key excerpts on "Deriving Equations"
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Chapter 6 Solving simple equations Why it is important to understand: Solving simple equations In mathematics, engineering and science, formulae are used to relate physical quantities to each other. They provide rules so that if we know the values of certain quantities, we can calculate the values of others. Equations occur in all branches of engineering. Simple equations always involve one unknown quantity which we try to find when we solve the equation. In reality, we all solve simple equations in our heads all the time without even noticing it. If, for example, you have bought two CDs, each for the same price, and a DVD, and know that you spent £25 in total and that the DVD was £11, then you actually solve the linear equation 2 x + 11 = 25 to find out that the price of each CD was £7. It is probably true to say that there is no branch of engineering, physics, economics, chemistry and computer science which does not require the solution of simple equations. The ability to solve simple equations is another stepping stone on the way to having confidence to handle engineering mathematics and science. At the end of this chapter, you should be able to: distinguish between an algebraic expression and an algebraic equation maintain the equality of a given equation whilst applying arithmetic operations solve linear equations in one unknown including those involving brackets and fractions form and solve linear equations involved with practical situations evaluate formulae by substitution of data 6.1 Introduction 3 x – 4 is an example of an algebraic expression. 3 x – 4 = 2 is an example of an algebraic equation (i.e. it contains an ‘=’ sign). An equation is simply a statement that two expressions are equal. Hence, A = π r 2 (where A is the area of a circle of radius r) F = 9 5 C + 32 (which relates Fahrenheit and Celsius temperatures) and y = 3 x + 2 (which is the equation of a straight line graph) are all examples of equations. Science and Mathematics for Engineering...
eBook - ePubMathematical Reasoning
Patterns, Problems, Conjectures, and Proofs
- Raymond Nickerson(Author)
- 2011(Publication Date)
- Psychology Press(Publisher)
...There are numerous views as to what constitutes the essence of mathematics, especially among mathematicians. A major purpose of this book is to explore some of those views and to catch a glimpse of what it means to reason mathematically.I suspect that for many people, mathematics is synonymous with computation or calculation. Doing mathematics, according to this conception, amounts to executing certain operations on numbers—addition, subtraction, multiplication, division. More complex mathematics might involve still other operations—raising a number to a specified power, finding thenth root of a number, finding a number’s prime factors, integrating a function.Computation is certainly an important aspect of mathematics and, for most of us, perhaps the aspect that has the greatest practical significance. Knowledge of how to perform the operations of basic arithmetic is what one needs in order to be able to make change, balance a checkbook, calculate the amount of a tip for service, make a budget, play cribbage, and so on. Moreover, the history of the development of computational techniques is an essential component of the story of how mathematics got to be what it is today. But, important as computation is, it plays a minor role, if any, in much of what serious mathematicians do when they are engaged in what they consider to be mathematical reasoning.Whatever else may be said about mathematics, even the casual observer will be struck by the rich diversity of the subject matter it subsumes. Ogilvy (1956/1984) suggests that mathematics can be roughly divided into four main branches—number theory, algebra, geometry, and analysis—but each of these major branches subsumes many subspecialties, each of which can be portioned into narrower subsubspecialties...
eBook - ePub- Wayne A. Wickelgren(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
...Recall that a critical aspect of solving many problems consists in retrieving from memory the relevant previously established relations and principles with common properties needed to solve the present problem. It may be that the current problem is really a special case of a general class of problems for which we already know a simple rule for solution. For example, if the present problem is the linear equation 2x + 5 = 13, we know that the solution to this particular linear equation can be achieved by using the general methods for solving any linear equation of the form ax + b = c. Similarly, if the equation were a quadratic of the form 7 x 2 + 2 x – 4 = 0, we have a formula for solving any equation of the form ax 2 + bx + c = 0. A broad range of higher order equations can be solved by certain types of numerical methods. If we have a particular equation that falls within the scope of a numerical method, we know we can apply this method to solve the particular problem. In a geometric context, if a problem gives two sides of a right triangle and we are asked to solve for the third side, we know a general method that is applicable to solving all such problems — namely, use of the Pythagorean Theorem, c 2 = a 2 + b 2. If we are given a problem in which we must determine the number of combinations of seven things taken four at a time, we need only retrieve the formula for the number of combinations of m things taken n at a time and substitute in the appropriate values for m and n in order to solve the problem. Ordinarily, to solve problems we must combine use of more than one previously established principle. Thus, in all proof problems, whether algebraic, geometric, or logical, the proof invariably requires a sequential application of several previously established principles...
