Mathematics
Rewriting Formulas and Equations
Rewriting formulas and equations involves manipulating mathematical expressions to isolate a specific variable or to solve for a different variable. This process often involves using inverse operations such as addition and subtraction, multiplication and division, and exponentiation and roots to rearrange the equation in a different form while preserving its equality.
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eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
4 MODULE Equations, word problems and manipulation of technical formulae 4.1. Linear equations On completion of this topic, you should be able to: 4.1.1 Solve linear equations without any fractions. 4. 2 Word problems On completion of this topic, you should be able to: 4.2.1 Set and solve linear equations from word problems (fractions in equations are excluded). 4.3 Manipulation of technical formulae On completion of this topic, you should be able to: 4.3.1 Change the subject of a given formula to any other subject. The following applications are excluded: i) Manipulation with exponents on a higher level than necessary for the Pythagorean theorem ii) Manipulation by factorisation iii) Manipulation by using the quadratic formula iv) Manipulation by using the laws of logarithms. 4.3.2 Determine the values of the new subject by substituting the values of the known quantities 4.3.3 Solve problems on distance, speed, time and revolutions. 112 Module 4 • Equations, word problems and manipulation of technical formulae 4.1. Linear equations Introduction An equation is a mathematical statement that includes an = sign. A linear equation is an equation in which the highest variable (unknown) you are solving for is 1. A linear equation cannot have more than one unique solution (often called root ) for x , for example: If 2 x + 1 = 9, the only solution that would make this statement true is if x = 4. Pre-knowledge • The four basic operations: addition; subtraction; multiplication and division • Simplification of brackets • Inverse operations of addition; subtraction; multiplication and division Inverse operations have an opposite effect on each other: • Addition and subtraction are inverse operations of each other. • Multiplication and division are inverse operations of each other. An equation can be represented by a balance scale, as the lefthand side (LHS) = the righthand side (RHS).
eBook - PDF- Dale Ewen(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
For more information, please visit www.cengage.com and access the Student Online Resources for this text. CHAPTER 6 OBJECTIVES ◆ Use the addition, subtraction, multiplication, and division properties of equality to solve simple equations. ◆ Solve equations with parentheses. ◆ Solve equations with fractions. ◆ Translate words into algebraic symbols. ◆ Solve application problems using equations. ◆ Solve a formula for a given letter. ◆ Substitute data into a formula and find the value of the indicated letter using the rules for working with measurements. ◆ Substitute data into a formula involving reciprocals and find the value of the indicated letter using a scientific calculator. Equations and Formulas sima/Shutterstock.com Diesel Technician Diesel technician repairing a diesel engine 222 CHAPTER 6 ◆ Equations and Formulas Equations 6.1 In technical work, the ability to use equations and formulas is essential. A variable is a sym-bol (usually a letter of the alphabet) used to represent an unknown number. An algebraic expression is a combination of numbers, variables, symbols for operations (plus, minus, times, divide), and symbols for grouping (parentheses or a fraction bar). Examples of alge-braic expressions are 4 x 2 9, 3 x 2 1 6 x 1 9, 5 x (6 x 1 4), 2 x 1 5 2 3 x An equation is a statement that two quantities are equal. The symbol “ 5 ” is read “equals” and separates an equation into two parts: the left member and the right member. For example, in the equation 2 x 1 3 5 11 the left member is 2 x 1 3 and the right member is 11. Other examples of equations are x 2 5 5 6, 3 x 5 12, 4 m 1 9 5 3 m 2 2, x 2 2 4 5 3( x 1 1) To solve an equation means to find what number or numbers can replace the variable to make the equation a true statement. In the equation 2 x 1 3 5 11, the solution is 4. That is, when x is replaced by 4, the resulting equation is a true statement.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
This algebraic equation gives the exact relationship between certain quantities, allowing you to find the value of a particular quantity, provided you know the values of all the other unknown quantities. To do this, the equation is manipulated in such a way that the variable you are looking for is isolated on the left hand side and everything else is written on the right hand side. We refer to this manipulation of putting the wanted variable on the left hand side as making it the subject of the formula . Many different professions use formulae to calculate various solutions. We will mainly deal with formulae used in the technical field. Pre-knowledge • Working with a scientific calculator • Solving algebraic equations such as those done in section 4.1 • Inverse operations Manipulation of technical formula is often described as changing the subject of the formula. R Rules for manipulating formulae Manipulating a formula is nothing more than solving for a variable. Remember the following. Operation Inverse operation Add Subtract Multiply Divide Square: x 2 Take the square root: x Raise to the power: x n Take the root of that power: x n Whatever you do to one side of the equation, you must also do to the other side to keep it balanced. NB: Do only one operation at a time. 121 Introductory Mathematics| Hands-On Examples Make the letter in brackets the subject of the formula. Problem Solution Explanation 1. a + b = c ( b ) a + b – a = c – a ∴ b = c – a • Use the inverse operation of addition (subtraction) to get rid of a on the LHS. 2. p – r = t ( p ) p – r + r = t + r ∴ p = t + r • Use the inverse operation of subtraction (addition) to get rid of r on the LHS. 3. A = l × b ( b ) l × b = A l b l × = A l ∴ b = A l • Rewrite the formula so that the variable you want to be the subject is on the LHS. • Use the inverse operation of multiplication (division) to get rid of l on the LHS.
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