# Multiple Equations In One Equation Latex

Step-by-step, we must get the variable out of parentheses, away from other terms, and with a coefficient of 1. In LaTeX, such a formula is typeset using the eqnarray* environment. But, it doesn't mean that 'solving for variables with only a single equation is always possible'. Those can be expressions as well. For example, in the following equation, it is preferable to ensure there is a decent amount of space between the maths and the text. Include terminal punctuation within the math environment. 1 Physical derivation Reference: Guenther & Lee §1. Equation Solver. For example, here are three essentially equivalent ways to code in LaTeX the same anti-derivative formula from calculus as an in-line equation. Just apply the utility by clicking Kutools > More > Convert Equation to Image. If I do it as you did, the text carries over to the next line, but the very start of the next line. \begin{eqnarray} Display multiple equations together or a long equa-tion that requires multiple lines. We find that the diagram helps the students represent the context in a way that leads to a correct equation. The above example was obtained by typing the lines. The Equation tool converts this into free form text which may be placed in whichever font you prefer. LaTeX doesn't break long equations to make them fit within the margins as it does with normal text. com is simply the right destination to have a look at!. However, most linear equations require more than one step in order to find their solution. Arial Times New Roman Symbol 1_Default Design Microsoft Equation 3. LaTeX, originally developed in the 1980s by Leslie Lamport, allows you to generate camera-ready format and equations using predefined macros rather than all manual text. 3x + 2y + 4z = 11 Add 2 times the second 4x º 2y + 6z = 8 equation to the first. The form ax = 0. A number of MCQ's, equations in geometry, translating two-step equations and many more exercises are available for practice. It is therefore up to you to format the equation appropriately (if they overrun the margin. In Algebra, each term within an equation is separated by a plus (+) sign, minus (-) sign or an equals sign (=). Matrices and other arrays in LaTeX. Solving a system of equations by addition is ideal when you see that both equations have one variable with the same coefficient with opposite charges. In fact, your example is probably best with the cases environment. ) This typically requires some creative use of an eqnarray to get elements shifted to a new line and to align nicely. A linear equation is an equation with two variables whose graph is a line. }The equation and align environment. unstack specifies that the individual equations from multiple-equation models (e. You can include mathematical expressions and equations in your document. This is done by adding the command \usepackage{Name} with the package name in the preamble. Revision worksheets, Sample papers, Question banks and easy to learn study notes for all classes and subjects based on CBSE andNCERT guidelines. Editor Online de Ecuaciones Latex is a Spanish version. Solving multiple equations numerically In the example above, we knew that there was only one intersection point since two lines can only intersect at a single point. Figures are visual presentations of results, such as graphs, diagrams, images, drawings, schematics, maps, etc. Below is a table with some common maths symbols. x – 3y = 8. I’m trying to recreate your example “multiple rows and columns”. Solving Absolute Value Equations Key Vocabulary absolute value equation, p. If you want to add a cross-reference to the entire block of equations, e. We tried to explain the trick of solving word problems for equations with two variables with an example. So, by saving some space, you're sacrificing some of that LaTeX functionality. That's the ideal scenario, anyway, but sometimes it becomes tricky. 5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. , integrals containing more than one integral sign) one finds that LaTeX puts too much space between the integral signs. It is therefore up to you to format the equation appropriately (if they overrun the margin. Iron is present in one reactant and one product, so balance its atoms first. Equation Solver. I have found the equation editor, which works fine. 0 format, scientific and mathematical constants database, physical science SI units database, interactive unit conversions, especially for students and teachers. Algebra Balance Equations Instructions Replicate the given equation by moving the blocks of X and 1 to the sides of the balance scale. This page produces high quality typeset images of equations from LaTeX equations. When a new equation is added to the document, the subsequent equations should be renumbered. Open an example of the amsmath package in Overleaf. The general appearance of quadratic equation is a second degree curve so that the degree power of one variable is twice of another variable. This equation assumes the use of creatinine methods with calibration traceable to IDMS. This is useful when you don't want to immediately compute an answer, or when you have a math "formula" to work on but don't know how to "process" it. Linear Equations and Inequalities in Two Variables Name_____ MULTIPLE CHOICE. If we want to have an equation stand out, we could use two di erent methods shown below: one is the fequation*genviornment and the other is a double. In the first case, mathematics mode is delimited by dollar signs. com gives both interesting and useful facts on online equation solver multiple unknowns, fractions and multiplication and other algebra topics. Linear equations math quiz questions and answers pdf, variables of linear equation is implicitly raised to, with answers for online certifications. Let us convert a short sample document containing both MathType and Microsoft Word equations. 5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. The common log function log(x) has the property that if log(c) = d then 10d = c. Question 1116108: Given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solution. In the replacement set, an identity equation is always satisfied by all the numbers that are present in the sets. If one equation has a variable isolated on one side, use the substitution method to substitute the expression for the isolated variable into the other equation. The next lines contain only the equations and annotations, I would like them aligned with the equation and annotation of the second line. If you want to extract the LaTeX out of a dead PDF, you can use Mathpix to take a screen shot of the equations in the PDF and it will automatically convert it to LaTeX, all you have to do is paste it into your editor. The new Office 2007 equation editor works in a different way than the old equation editor. This approach also creates a label, which we can refer to later if we like. It is possible to convert all equations in a document to the Professional or Linear formats, or a single equation only, if the math zone is selected or the cursor is in the equation. In this example, you use the first equation in the original system, which now becomes x + 2(3) + 3(–4) = –7. Yes You can easily write maths equations in OneNote. Also, when choosing multiple-choice questions you can add equations in the possible answers. On this page, you will find Algebra worksheets mostly for middle school students on algebra topics such as algebraic expressions, equations and graphing functions. Multiplying (or dividing) the expression on each side by the same number does not alter the equation. \begin{eqnarray} Display multiple equations together or a long equa-tion that requires multiple lines. In the opening Equations dialog box, click the Yes button. The final result of solving the equation is simplified so it could be in a different form than what you expect. Simple fractional equations. Next we solve for y by substituting for z in the equation formed by the second row: Finally we solve for x by substituting the values of y and z in the equation formed by the first row: Therefore, the solution to the system of equations is {x,y,z} = {1,-2,1}. Solve for the variable. Consider the problem of typesetting the formula It is necessary to ensure that the = signs are aligned with one another. 1 To solve equations TOP: 1-4 Problem 2 Solving a Multi-Step Equation. We can simplify both equations by multiplying each separate one by it’s LCD, just like you can do when you are working with one equation. Equations with Multiple Variables So far, when I've asked you to solve an equation for a variable, it was pretty obvious which one I was talking about. Here we see how to create an equation using the \begin{equation} and \end{equation} commands. No highly advanced LaTeX tools such as graphing, commutative diagrams, or geometric shapes are implemented. Create a new equation of the same type. Using variables. These images can be used on web pages or in other documents. This command forces LaTeX to give an equation the full height it needs to display as if it were on its own line. The equations of a system are dependent if ALL the solutions of one equation are also solutions of the other two equations. Khan Academy is a 501(c)(3) nonprofit organization. Diophantine equations are important when a problem requires a solution in whole amounts. How do you solve for multiple variables in excel? I am trying to write a program to evaluate Real Estate value. To add an equation, you compose it in the Pages equation dialog using LaTeX commands or MathML elements, then insert it into your document. Contents 1 Introduction 2 Including the amsmath package 3 Writing a single equation 4 Displaying long equations. 6 Math Perhaps the most important function of Latex is its ability to easily print equations. Please specify the options in the Batch Insert Captions dialog box. Multiple devices with one IPv6 to the Internet?. For that kind of equations, select the LaTeX equation type and type your equation within \begin{} and \end{} instructions. An equation is a mathematical statement that two things are equal. MathMagic is a WYSIWYG math editor with Graphic user interface, with support for MathML, LaTeX, MS Equation Editor, and more. When numbering is allowed, you can label each row individually. The version of LaTeX used is a subset of AMS-LaTeX markup, a superset of LaTeX markup which is in turn a superset of TeX markup, for mathematical formulae. Multiple Lines and Multiple Equations As previously mentioned, Origin's LaTeX App wraps user-entered equation markup in a single equation environment by automatically adding \begin{equation} and \end{equation} commands before and after user-entered markup. It is the most used markup for equations. In some multiple integrals (i. LaTeX forum ⇒ Math & Science ⇒ Vertical spacing of multiple equations Information and discussion about LaTeX's math and science related features (e. The technique of clearing fractions for solving linear equations. Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. Exercises on the application of the equations in real life are available here to impart practical knowledge. The “heart” of solving a literal equation is to isolate or keep by itself a certain variable on one side of the equation (either left or right) and the rest on the opposite side. In the replacement set, an identity equation is always satisfied by all the numbers that are present in the sets. I would like to find say 20 variations of x, u and v which will add up to this solution, within certain boundary constraints for x, u and v. LaTeX assumes that each equation consists of two parts separated by a &; also that each equation is separated from the one before by an &. To start practicing, just click on any link. Move the equations to a full-width figure, where you can play all of the same games. Equations with Multiple Variables So far, when I've asked you to solve an equation for a variable, it was pretty obvious which one I was talking about. Using the split environment within an align will produce one equation number, vertically centered within the split (so long as there's not a page break involved, as discussed here). Multi-step equation worksheets. Hello, I have an equation like this one: f(x)=1+g(x) where g(x) is a very long term, and I have to split it into two halfs, but I want the second Multi-line Equation Problem in Latex | Physics Forums. This is done by adding the command \usepackage{Name} with the package name in the preamble. Multiple Definitions. Writing Equations with LaTeX Print If your field requires you to use equations regularly, and especially if you plan to write a paper or publish an article or book that includes multiple and complex equations, the computer language that you are likely to have to use is called LaTeX. Examples here are matrices, multiple aligned equations, and binomial coefficients. Multiple Representations: Verbal to Equations Teacher2Teacher Grade 8 Teacher: Yes, the diagrams help us write equations for contexts. Download CBSE Class 8 Maths - Linear Equations in One Variable MCQs in pdf, Chapter wise Multiple Choice Questions (MCQs) for Linear Equations, CBSE Class 8 Maths - Linear Equations in One Variable MCQs. Display math mode (equations are separated from the text) • Most math symbols can only be used in math mode (e. Algebra Four: Students play a generalized version of connect four, gaining the chance to place a piece on the board by solving an algebraic equation. Let us convert a short sample document containing both MathType and Microsoft Word equations. If eqn is a symbolic expression (without the right side), the solver assumes that the right side is 0, and solves the equation eqn == 0. LaTeX math. I did for 2 equations but not for 3. In the Applications/Utilities folder, there is a program called Grapher. Equations like that in the next example occur frequently in applications. Notice: Undefined index: HTTP_REFERER in /home/yq2sw6g6/loja. Then one way to write the. Section 5-4 : Systems of Differential Equations. Using Augmented Matrices to Solve Systems of Linear Equations 1. Type the following: The first equation x+y=7. This lesson covers inserting a preset equation and also manually entering an equation. Break the equation across two lines: this works up to about 1. Solving Equations with e and ln x We know that the natural log function ln(x) is deﬁned so that if ln(a) = b then eb = a. ) Graham Heywood / istockphoto. Sometimes not well, but at least the process is a vaguely familiar one. I have an extremely long equation that does not fit into a single line. Open an example of the amsmath package in Overleaf. As you can see, this method only required one Maple command to get both the and the coordinates. If you use images instead, you cannot change the equations without editing your images. Linear equations in one variable may take the form $ax+b=0$ and are solved using basic algebraic operations. The value of a counter can be changed with a command of the type \setcounter{equation}{0}. Given two equations and two unknowns, our objective is to reduce this to one equation and one unknown, which we know we can solve. How do I reference my LaTeX tables or equations? Related articles. These linear systems can have many variables, and you can solve those systems as long as you have one unique equation per variable. No highly advanced LaTeX tools such as graphing, commutative diagrams, or geometric shapes are implemented. One click to add right aligned captions for all tables, equations, figures, charts, and SmartArts in Word. When we are given this type of equations, we are always moving variables to the one side, and real numbers to the other side of the equals sign. order equations. equations are done one with equationand one with eqnarray. And the only real solutions are x=0,y=1. Interactive Open Source Numerical Library of C/C++ functions for Engineering, Finance, Maths, Science and Statistics. It looks like a fraction surrounded by parentheses, but without the line in the middle. I will then ask students to brainstorm why multiple methods works, and to decide if eliminating one path was preferable to another. In mathematics, an equation is a statement that asserts the equality of two expressions. Unicode math. Include terminal punctuation within the math environment. The relation operator == defines symbolic equations. Is possible to compute margins condition to more than one equation in cmp? are there other solutions? (code and questions are below and in attachment ). Can I write a LaTeX equation over multiple lines? Using the multiline, aligned packages. where a and b are some real numbers. k Worksheet by Kuta Software LLC. If an equation is longer than one line or several formulas must be. Click here to see ALL problems on Quadratic Equations Question 1142926 : If one throws a ball down from a high cliff, the distance it travels can be modeled by the equation: LaTeX: d = -9. Age Problems - Equations in single variable - Algebra This worksheet is based on age word problems which come under equations in one variable. Write one equation above the other. LaTeX numbering One advantage of LaTeX over the other TeX-flavors is that it provides an automatic numbering of the sections, theorems, equations etc. How to Solve the System of Equations in Algebra Calculator. Exclude equations from numbering. A value of the variable which makes the two sides of the equation equal is called the solution of the […]. Numbering Equations. mdl) file in a separate text editor. For example, let us eliminate z. Step-by-step, we must get the variable out of parentheses, away from other terms, and with a coefficient of 1. But I have pages with many equations. The “heart” of solving a literal equation is to isolate or keep by itself a certain variable on one side of the equation (either left or right) and the rest on the opposite side. Exercises on the application of the equations in real life are available here to impart practical knowledge. Step 2 Add the equations obtained in Step 1. The linear option will display the equation in either UnicodeMath format, or LaTeX format, which can be set in the Conversions chunk. Linear Systems: Using Graphs & Tables Teacher Notes Objectives To illustrate how to locate the real number solution of a system of linear equations (two equations and two variables) using tables. Exercises 3–11 y22 y1 01 3 45678 Graph y2. If you want to typeset equations beyond the very simplest ones, you will want to use the amsmath package, one of the many packages that you can use with LaTeX. In LaTeX we can label equations for easy reference within the article. Note that math mode ignores whitespace, in fact, this whole code could have been put on one line and still would have compiled correctly. If you have questions or problems with this page please email me. Using Inverse Operations to Solve Equations: Overview. Use the addition, subtraction, multiplication, and division properties of equalities to solve linear equations. For example, to solve the equation 3 x + 2 = 23 , you'd solve for (isolate) the x variable. That means that within systems of linear equations you have two or more linear equations with the same variables. If an equation is longer than one line or several formulas must be. Managed to solve this using Excel Solver but it only returns one solution and I would like several. Hancock Fall 2006 1 The 1-D Heat Equation 1. The equations must contain a comparison character such as equal, ie. This command forces LaTeX to give an equation the full height it needs to display as if it were on its own line. Let's solve one simple example of a linear equation with one variable: 4x - 2 = 2x + 6. For small amplitudes, the period of such a pendulum can be approximated by:. The Export Equations command allows you to export MathType and Equation Editor equations in your Word document to individual graphics files. When combined with adjusting font size, you can often adjust where the break occurs to make it look reasonable. (thanks to umustbe and thornahawk for the links). The desire was to have the equation be centered on the page, and the equation's number appear at the right margin within brackets. To show LaTex equation in Axis Title and Legend Add LaTeX equations to worksheet metadata area (e. Output equation:\text{This formula }f(x) = x^2\text{ is an example. 1: Changing the order in which the equations are listed produces an equivalent system. "And answer to that is no. Other features include the ability to reload a equation image and make changes and re upload to your presentation. How Office 2007's equation editor "builds-up" equations. The example of a scale where we try to achieve balance helps to explain why we can subtract one equation from the other in a system of equations. That additional step may be something like multiplying the variable by a certain number to get rid of a fraction in front of it. By default, vpasolve returns the same solution on every call. How do I convert my LaTeX document to Word? Sometimes a long equation needs to be broken over multiple lines, especially if using a double column export style. This step uses the techniques used when solving a linear equation in one variable. The incompressible Navier-Stokes equation with mass continuity (four equations in four unknowns) can be reduced to a single equation with a single dependent variable in 2D, or one vector equation in 3D. I had tried using the \subequation environment - However, this environment seems to tag each equation with ascending letters (1a), (1b), etc. I have 2 questions: One- Is it possible to place two figures side by side and immediately below them another two figures, all of these as one “unit”?. On this page, you will find Algebra worksheets mostly for middle school students on algebra topics such as algebraic expressions, equations and graphing functions. The new Office 2007 equation editor works in a different way than the old equation editor. A sinusoidal plane wave is one special solution of these equations. Output equation: $\text{This formula }f(x) = x^2\text{ is an example. The equation names have to be separated by commas or by a line break. \begin{equation} math \end{equation} to display mathematics. When numbering is allowed, you can label each row individually. "100x" -> "100x", add some input validation, in particular check whether the equation is actually linear and not quadratic or cubic, and finally add a GUI to solve and plot multiple linear functions using different colors and get a nice tool for use in elementary mathematical education. Here we arrange the equations in three columns. A linear equation is an equation of a straight line, written in one variable. I did for 2 equations but not for 3. LaTeX assumes that each equation consists of two parts separated by a &; also that each equation is separated from the one before by an &. The deflection due to all loads are v B-load and v C-load. How do I use Excel to plug multiple numbers into a single equation and then give the results for each? I'm simplifying the below example to make it easier. Improve your math knowledge with free questions in "Solve linear equations" and thousands of other math skills. Now that we have solved equations in one variable, we will now work on solving equations in two variables and graphing equations on the coordinate plane. See screenshot:. Multiple choice test Equivalence of equations. The reason is that equation numbers are there for anyone to refer to and not just you the author. dCode can solve equations and find variables. Write beautiful math equations & symbols easily in MathMagic, use them widely in your word processors, Presentations, DTP software. This step uses the techniques used when solving a linear equation in one variable. So, for any equation an absolute value, there is at least one turning point value of the variable for each absolute value. AMS-LaTeX is a set of LaTeX packages for mathematics, which was developed by the American Mathematical Society so that mathematicians could produce beautiful mathematics in LaTeX, mathematics acceptible for publication in the society's journals. Connecting graphs, tables, and equations of lines is an important practice so that we can to help understand lines and how to graph them. Contents 1 Introduction 2 Including the amsmath package 3 Writing a single equation 4 Displaying long equations. Repeat until the system is reduced to a single linear equation. You might know what equations you want to refer to but you don't know in advance what equations your readers. Still confusing with Word, MathType, LaTeX, Lyx, TexMacs? Let's try AxMath! Download it now! Link2. Know what a linear equation is.$\endgroup\$ - A. The changing magnetic field creates a changing electric field through Faraday's law. That's the ideal scenario, anyway, but sometimes it becomes tricky. com for the archive. Step 2 Add the equations obtained in Step 1. The next lines contain only the equations and annotations, I would like them aligned with the equation and annotation of the second line. The Linear Equations Worksheets are randomly created and will never repeat so you have an endless supply of quality Linear Equations Worksheets to use in the classroom or at home. To do this: 1. The new equation editor is completely unusable. To solve an equation, we transform it into simpler equivalent equations until we can easily read off the value(s) of the variable that make the equation true. Remember, the point-slope formula is only one type of linear equation. In the first case, mathematics mode is delimited by dollar signs. When numbering is allowed, you can label each row individually. More than one equation may be declared at once. (this means that the highest power of the function, u, and its derivatives is just equal to one in each term in the equation, and that only one of them appears in each term). It is therefore up to you to format the equation appropriately (if they overrun the margin. "100x" -> "100x", add some input validation, in particular check whether the equation is actually linear and not quadratic or cubic, and finally add a GUI to solve and plot multiple linear functions using different colors and get a nice tool for use in elementary mathematical education. Step 3: The results from steps one and two will each be an equation in two variables. Integration and Compatibility. For example,. How do I do it? Thanks. Step 2 If necessary, multiply either equation or both equations by appropriate numbers so that the sum of the orthe sum of the is 0. Is possible to compute margins condition to more than one equation in cmp? are there other solutions? (code and questions are below and in attachment ). The equation in one variable, together with either of the original equations, then forms an equivalent system whose solution is easily obtained. Create a personal Equation Sheet from a large database of science and math equations including constants, symbols, and SI units. Again, use * to toggle the equation numbering. Hello, I have an equation like this one: f(x)=1+g(x) where g(x) is a very long term, and I have to split it into two halfs, but I want the second Multi-line Equation Problem in Latex | Physics Forums. Practice: Identify equations from visual models (tape diagrams) Practice: Identify equations from visual models (hanger diagrams) So once again, we have three equal, or we say three identical objects. In the Applications/Utilities folder, there is a program called Grapher. He was asking about inline equations. By doing it this way it is possible to then change the equation in the LaTeX-editor, compile and then only an update of the figure-link in InDesign is needed. 1 Maths Environments. LaTeX / public domain. 15 Resolved Linear Equations. \begin{eqnarray} Display multiple equations together or a long equa-tion that requires multiple lines. How can one write a long mathematical equation in latex? I think the use of multiple lines structured environment can work. OneNote 2010 provides a very convenient way to write-in mathematical equations. It is a method of solving linear system of equations. electronic journal of differential equations (ejde) Since its foundation in 1993, this e-journal has been dedicated to the rapid dissemination of high quality research in mathematics. This yields a system of equations with one fewer equation and one fewer unknown. First, select the equation – here, drag through the equation, omitting the words (i. , integrals containing more than one integral sign) one finds that LaTeX puts too much space between the integral signs. In LaTeX that is no longer true, but it is true that the size of these units will scale with the font, so that a change to a different font size will not disturb the proportions of the typeset material. Here we see how to create an equation using the \begin{equation} and \end{equation} commands. This post explains how to typeset multi-part definitions in LaTeX. It supports polynomial equations as well as some equations with exponents, logarithms and trigonometric functions. By default, vpasolve returns the same solution on every call. Long Name, Units, Comments, etc. The multiline environment lets you split equations over multiple lines, aligning the beginning of the equation with the left margin, and the end of it with the right margin. There are two linear formats for math that Word supports:. To number equations, we use the align environment. Using standard cross-referencing in LaTeX only produces the label number, a name describing the label such as figure, chapter or equation has to be added manually. Solving an Algebraic Linear Equation with One Variable. Solution to Equation 6: 2 log 3 (2x) = log 3 (4) + log 3 (3 + 2x) There are two things to notice in this equation: both sides have logarithms, and one side has multiple logarithms. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English any equality is an equation. But to answer your question directly, used the aligned environment within equation - this way the outside environment gives the number:. one has to specify which packages are going to be used in the document. Actually the answer to the original question 'Is it possible to solve for two variables related by single equation' is YES !. This example shows a linear system with two equations and two variables. Equations with Multiple Variables So far, when I've asked you to solve an equation for a variable, it was pretty obvious which one I was talking about. format) or colon (European format). Previous ones: Basics and overview Use of mathematical symbols in formulas and equations Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. This makes vpasolve use a random initial guess which can lead to different solutions on successive calls. Note that math mode ignores whitespace, in fact, this whole code could have been put on one line and still would have compiled correctly. We will multiply equation. Hello all! I'm writing my first real LaTeX document and I have just one issue I haven't been able to figure out. To insert equations in Word from one of the preset equations, first place your cursor at the insertion point in your document where you want the equation to appear. It would be inconvenient to have to exit math mode to type the text in normal roman type, so the most common function names are defined as special commands. Open a Google Docs document, click on "Insert" and select "Equation" from the menu. To write an equation in general linear form, given a graph of the equation, first find the x-intercept and the y-intercept -- these will be of the form (a, 0) and (0, b). In the applet below, lines can be dragged as a whole or with one of the two defining points. In this post, we will talk about separable. Linear equations math multiple choice questions and answers (MCQs), linear equations math quiz answers pdf to learn business math online courses. Type the equation you want to calculate. One can enter the equation in LaTeX markup, and the formatted equation is inserted as an object. No LaTeX typesetting tools such as labels and references are implemented. Use either. = (or or >).