About Mathematical Equations

Transcription

1 About Mathematical Equations

2 TABLE OF CONTENTS About Mathematical Equations... 1 Solving a Mathematical Equation... 1 Rules for Rearranging Equations... 2 Rule # Rule # Order of Operations... 3 Pairs of Inverse Operations... 3 Rearranging Equations... 4 Chemistry Applications... 4 Using Formulas to Solve for Unknown Variables... 5 Glossary... 6 References... 7

3 About Mathematical Equations A mathematical equation is a statement that illustrates the relationship between two or more variables using algebraic operations (i.e., addition, subtraction, multiplication, division, etc.). An equation contains two expressions separated by an equal sign, where the expression on the left-hand side of the equal sign is equivalent to the expression on the right-hand side. A noticeable difference between equations (or sometimes called equalities) and inequalities is that an equation contain an equal sign (=), whereas inequalities contain a greater than (>) or a less than (<) sign. Solving a Mathematical Equation Solving a mathematical equation implies determining its solution(s). Most often, an equation may contain a set of variables in which all variables, except for one, are known or given. On the other hand, an equation may also contain more than one unknown variable. In this case, a system of equations (i.e., simultaneous equations; multiple linear equations) must be involved in order to solve for the multiple unknown variables. Regardless of the situation, rearranging the equation(s) is necessary. Aside: In this module, the topic of discussion will only pertain to equations with one unknown variable. Systems of equations as well as solving for multiple-unknown variables will not be discussed. 1

4 Learning how to rearrange an equation properly is critical for success in courses that regularly use formulas. Some of these courses may be biology, chemistry, physics, economics, and calculus. Here are some reasons why you should develop your ability to rearrange equations: Rule #1 Equations are easier to work with before inserting numbers. It is always best to isolate the unknown variable on one side of the equal sign before inserting the numerical values into the equation. After learning how to rearrange equations, you will only have to memorize one general equation containing the variables of interest. Any other variables within the equation can be solved by rearranging the general equation. This will reduce the amount of memorization on your part, and will ultimately save you valuable time! Rules for Rearranging Equations When performing an algebraic operation on one side of the equation, you must perform the same algebraic operation on the other side of the equation. Rule #2 To eliminate a variable or a constant from one side of the equation, you must perform the inverse algebraic operation on both sides of the equation. NOTE: In addition to these two rules, you will also need to recognize two very important things: 1) The order of operations in which the expression should be evaluated 2) The pairs of algebraic operations that are opposite to each other (i.e., an operation and its inverse operation) 2

5 Order of Operations When evaluating an algebraic expression, a specific order is associated with carrying out the calculations --- this is called the order of operations. The order of operations is as follows: Brackets and Parentheses (1 st priority) Exponents Division Multiplication Addition Subtraction (2 nd priority) (3 rd priority) (4 th priority) *Mnemonics BEDMAS (5 th priority) (6 th priority) Note: In an expression that involves two or more operations of the same priority level, perform the algebraic operations from left to right. Pairs of Inverse Operations Two algebraic operations are inverses of one another if they cancel each other out. Addition Subtraction Multiplication Division [adding and subtracting the same number does not change the original number] [multiplying and dividing the same number does not change the original number] 3

6 Rearranging Equations There are several general steps to follow when a question requires you to manipulate an equation in order to solve for an unknown variable: 1) Examine the equation that you are given ask yourself: which variables are known and unknown? 2) Identify the unknown variable that you are asked to solve 3) Rearrange the equation such that the unknown variable is isolated on one side of the equation, while all the other known variables are on the other side (note: you must follow the two rules for rearranging equations) 4) Replace the known variables with their corresponding values 5) Evaluate and simplify the equation to determine the value of the unknown variable Chemistry Applications In chemistry, you will encounter many different relationships, including the ideal gas law, equilibrium-constant expression, rate law, Gibbs Free Energy, and the Nernst equation. Because there are so many different formulas, with several different symbols, and thus several types of variables, one of the most important things to know is what each variable stands for in the formula. Substituting the correct values with the appropriate units in the correct places within the formula is extremely critical. It is important to note that when replacing a variable with a value, its corresponding units need to be incorporated as well. If all the variables are properly substituted, the units will be consistent in all parts of the formula. To illustrate this point, consider the ideal gas law: P V = n R T P = pressure V = volume n = number of moles R = molar gas constant T = temperature 4

7 Using Formulas to Solve for Unknown Variables Sometimes, in order to determine an unknown variable, you will need to algebraically rearrange the formula such that the unknown variable is isolated on one side of the equation. Take this formula as an example: ΔG = -n F E If you are asked to solve for ΔG (given the values for E, n, F), you can easily plug in the known values (i.e., n, F, E ). Once the values and their appropriate units are substituted, you can carry out the calculations directly. However, if you are asked to solve for E (given the values for ΔG, n, F), you will have to algebraically manipulate the formula such that E is isolated on one side of the equation. For instance: ΔG = -n F E divide both sides by n F [ ΔG / -n F ] = [ -n F E / -n F ] [ ΔG / -n F ] = E or E = [ ΔG / -n F ] Now, you can plug in the known values of ΔG, n, F, and perform the mathematical calculations. Formulas may get more complicated. Under such circumstances, it is important for you to be extra careful with the manipulation of the formulas. One method to ensure that you are on the right track is to pay attention to the units. Make certain that the unit of your final answer is appropriate for the quantity you are calculating. If you notice that the unit is inappropriate, then there is probably something wrong with the manipulation. 5

8 Glossary Inverse operations: Mathematical equation: Order of operations: System of equations: two algebraic operations are inverses of one another if they cancel each other out. a statement that illustrates the relationship between two or more variables using algebraic operations (i.e., addition, subtraction, multiplication, division, etc.). a specific order associated with carrying out the calculations when evaluating an algebraic expression. a set of linear equations involving the same set of variables (also known as simultaneous equations). 6

9 References "Equations." Math Study Guide. July 2008 < "Mathematical Formula." Massey University. July 2008 < "Rearranging Equations." Introductory Geoscience. July 2008 < 7