Jurisprudence      01/12/2024

How to solve equation m. Solve equations with fractions online. Solving even more complex linear equations

The online equation solving service will help you solve any equation. Using our website, you will receive not just the answer to the equation, but also see a detailed solution, that is, a step-by-step display of the process of obtaining the result. Our service will be useful to high school students and their parents. Students will be able to prepare for tests and exams, test their knowledge, and parents will be able to monitor the solution of mathematical equations by their children. The ability to solve equations is a mandatory requirement for schoolchildren. The service will help you educate yourself and improve your knowledge in the field of mathematical equations. With its help, you can solve any equation: quadratic, cubic, irrational, trigonometric, etc. The benefits of the online service are priceless, because in addition to the correct answer, you receive a detailed solution to each equation. Benefits of solving equations online. You can solve any equation online on our website absolutely free. The service is completely automatic, you don’t have to install anything on your computer, you just need to enter the data and the program will give you a solution. Any errors in calculations or typos are excluded. With us, solving any equation online is very easy, so be sure to use our site to solve any kind of equations. You only need to enter the data and the calculation will be completed in a matter of seconds. The program works independently, without human intervention, and you receive an accurate and detailed answer. Solution of the equation in general form. In such an equation, the variable coefficients and the desired roots are interconnected. The highest power of a variable determines the order of such an equation. Based on this, various methods and theorems are used for equations to find solutions. Solving equations of this type means finding the required roots in general form. Our service allows you to solve even the most complex algebraic equation online. You can obtain both a general solution to the equation and a particular one for the numerical values ​​of the coefficients you specify. To solve an algebraic equation on the website, it is enough to correctly fill out only two fields: the left and right sides of the given equation. Algebraic equations with variable coefficients have an infinite number of solutions, and by setting certain conditions, partial ones are selected from the set of solutions. Quadratic equation. The quadratic equation has the form ax^2+bx+c=0 for a>0. Solving quadratic equations involves finding the values ​​of x at which the equality ax^2+bx+c=0 holds. To do this, find the discriminant value using the formula D=b^2-4ac. If the discriminant is less than zero, then the equation has no real roots (the roots are from the field of complex numbers), if it is equal to zero, then the equation has one real root, and if the discriminant is greater than zero, then the equation has two real roots, which are found by the formula: D = -b+-sqrt/2a. To solve a quadratic equation online, you just need to enter the coefficients of the equation (integers, fractions or decimals). If there are subtraction signs in an equation, you must put a minus sign in front of the corresponding terms of the equation. You can solve a quadratic equation online depending on the parameter, that is, the variables in the coefficients of the equation. Our online service for finding general solutions copes well with this task. Linear equations. To solve linear equations (or systems of equations), four main methods are used in practice. We will describe each method in detail. Substitution method. Solving equations using the substitution method requires expressing one variable in terms of the others. After this, the expression is substituted into other equations of the system. Hence the name of the solution method, that is, instead of a variable, its expression is substituted through the remaining variables. In practice, the method requires complex calculations, although it is easy to understand, so solving such an equation online will help save time and make calculations easier. You just need to indicate the number of unknowns in the equation and fill in the data from the linear equations, then the service will make the calculation. Gauss method. The method is based on the simplest transformations of the system in order to arrive at an equivalent triangular system. From it, the unknowns are determined one by one. In practice, you need to solve such an equation online with a detailed description, thanks to which you will have a good understanding of the Gaussian method for solving systems of linear equations. Write down the system of linear equations in the correct format and take into account the number of unknowns in order to accurately solve the system. Cramer's method. This method solves systems of equations in cases where the system has a unique solution. The main mathematical action here is the calculation of matrix determinants. Solving equations using the Cramer method is carried out online, you receive the result instantly with a complete and detailed description. It is enough just to fill the system with coefficients and select the number of unknown variables. Matrix method. This method consists of collecting the coefficients of the unknowns in matrix A, the unknowns in column X, and the free terms in column B. Thus, the system of linear equations is reduced to a matrix equation of the form AxX=B. This equation has a unique solution only if the determinant of matrix A is different from zero, otherwise the system has no solutions, or an infinite number of solutions. Solving equations using the matrix method involves finding the inverse matrix A.

Application

Solving any type of equations online on the site for students and schoolchildren to consolidate the studied material. Solving equations online. Equations online. There are algebraic, parametric, transcendental, functional, differential and other types of equations. Some classes of equations have analytical solutions, which are convenient because they not only give the exact value of the root, but also allow you to write the solution in the form of a formula, which may include parameters. Analytical expressions allow not only to calculate the roots, but also to analyze their existence and their quantity depending on the parameter values, which is often even more important for practical use than the specific values ​​of the roots. Solving equations online.. Equations online. Solving an equation is the task of finding such values ​​of the arguments at which this equality is achieved. Additional conditions (integer, real, etc.) can be imposed on the possible values ​​of the arguments. Solving equations online.. Equations online. You can solve the equation online instantly and with high accuracy of the result. The arguments to specified functions (sometimes called "variables") are called "unknowns" in the case of an equation. The values ​​of the unknowns at which this equality is achieved are called solutions or roots of this equation. The roots are said to satisfy this equation. Solving an equation online means finding the set of all its solutions (roots) or proving that there are no roots. Solving equations online.. Equations online. Equations whose sets of roots coincide are called equivalent or equal. Equations that do not have roots are also considered equivalent. The equivalence of equations has the property of symmetry: if one equation is equivalent to another, then the second equation is equivalent to the first. The equivalence of equations has the property of transitivity: if one equation is equivalent to another, and the second is equivalent to a third, then the first equation is equivalent to the third. The equivalence property of equations allows us to carry out transformations with them, on which methods for solving them are based. Solving equations online.. Equations online. The site will allow you to solve the equation online. Equations for which analytical solutions are known include algebraic equations of no higher than the fourth degree: linear equation, quadratic equation, cubic equation and equation of the fourth degree. Algebraic equations of higher degrees in the general case do not have an analytical solution, although some of them can be reduced to equations of lower degrees. Equations that include transcendental functions are called transcendental. Among them, analytical solutions are known for some trigonometric equations, since the zeros of trigonometric functions are well known. In the general case, when an analytical solution cannot be found, numerical methods are used. Numerical methods do not provide an exact solution, but only allow one to narrow the interval in which the root lies to a certain predetermined value. Solving equations online.. Equations online.. Instead of an equation online, we will imagine how the same expression forms a linear relationship, not only along a straight tangent, but also at the very point of inflection of the graph. This method is indispensable at all times in the study of the subject. It often happens that solving equations approaches the final value by using infinite numbers and writing vectors. It is necessary to check the initial data and this is the essence of the task. Otherwise, the local condition is converted to a formula. Inversion in a straight line from a given function, which the equation calculator will calculate without much delay in execution, the offset will serve as a privilege of space. We will talk about students' success in the scientific environment. However, like all of the above, it will help us in the process of finding and when you solve the equation completely, store the resulting answer at the ends of the straight line segment. Lines in space intersect at a point and this point is called intersected by the lines. The interval on the line is indicated as previously specified. The highest post for the study of mathematics will be published. Assigning an argument value from a parametrically specified surface and solving the equation online will be able to outline the principles of productive access to a function. The Möbius strip, or infinity as it is called, looks like a figure eight. This is a one-sided surface, not two-sided. According to the principle generally known to everyone, we will objectively accept linear equations as the basic designation as it is in the field of research. Only two values ​​of sequentially given arguments are able to reveal the direction of the vector. Assuming that another solution to online equations is much more than just solving it means obtaining a full-fledged version of the invariant as a result. Without an integrated approach, it is difficult for students to learn this material. As before, for each special case, our convenient and smart online equation calculator will help everyone in difficult times, because you just need to specify the input parameters and the system itself will calculate the answer. Before we start entering data, we will need an input tool, which can be done without much difficulty. The number of each answer estimate will lead to a quadratic equation to our conclusions, but this is not so easy to do, because it is easy to prove the opposite. The theory, due to its characteristics, is not supported by practical knowledge. Seeing a fraction calculator at the stage of publishing the answer is not an easy task in mathematics, since the alternative of writing a number on a set helps to increase the growth of the function. However, it would be incorrect not to talk about student training, so we will each say as much as it needs to be done. The previously found cubic equation will rightfully belong to the domain of definition and contain the space of numerical values, as well as symbolic variables. Having learned or memorized the theorem, our students will show themselves only at their best, and we will be happy for them. Unlike multiple field intersections, our online equations are described by a plane of motion by multiplying two and three numerical combined lines. A set in mathematics is not defined uniquely. The best solution, according to students, is a complete recording of the expression. As was said in scientific language, the abstraction of symbolic expressions does not enter into the state of affairs, but the solution of equations gives an unambiguous result in all known cases. The duration of the teacher's lesson depends on the needs for this proposal. The analysis showed the necessity of all computational techniques in many areas, and it is absolutely clear that an equation calculator is an indispensable tool in the gifted hands of a student. A loyal approach to the study of mathematics determines the importance of views from different directions. You want to identify one of the key theorems and solve the equation in such a way, depending on the answer of which there will be a further need for its application. Analytics in this area is gaining momentum. Let's start from the beginning and derive the formula. Having broken through the level of increase of the function, the line along the tangent at the inflection point will certainly lead to the fact that solving the equation online will be one of the main aspects in constructing that same graph from the argument of the function. An amateur approach has the right to be applied if this condition does not contradict the conclusions of the students. It is the subtask that puts the analysis of mathematical conditions as linear equations in the existing domain of definition of the object that is brought into the background. Netting in the direction of orthogonality cancels out the advantage of a single absolute value. Modulo solving equations online gives the same number of solutions if you open the brackets first with a plus sign and then with a minus sign. In this case, there will be twice as many solutions, and the result will be more accurate. A stable and correct online equation calculator is success in achieving the intended goal in the task set by the teacher. It seems possible to choose the right method due to the significant differences in the views of great scientists. The resulting quadratic equation describes the curve of lines, the so-called parabola, and the sign will determine its convexity in the square coordinate system. From the equation we obtain both the discriminant and the roots themselves according to Vieta’s theorem. The first step is to represent the expression as a proper or improper fraction and use a fraction calculator. Depending on this, the plan for our further calculations will be formed. Mathematics with a theoretical approach will be useful at every stage. We will definitely present the result as a cubic equation, because we will hide its roots in this expression in order to simplify the task for a student at a university. Any methods are good if they are suitable for superficial analysis. Extra arithmetic operations will not lead to calculation errors. Determines the answer with a given accuracy. Using the solution of equations, let's face it - finding the independent variable of a given function is not so easy, especially during the period of studying parallel lines at infinity. In view of the exception, the need is very obvious. The polarity difference is clear. From the experience of teaching at institutes, our teacher learned the main lesson in which online equations were studied in the full mathematical sense. Here we were talking about higher efforts and special skills in applying the theory. In favor of our conclusions, one should not look through a prism. Until recently, it was believed that a closed set rapidly increases over the region as it is and the solution of the equations simply needs to be investigated. At the first stage, we did not consider all possible options, but this approach is more justified than ever. Extra actions with brackets justify some advances along the ordinate and abscissa axes, which cannot be missed with the naked eye. In the sense of an extensive proportional increase in the function, there is an inflection point. Once again we will prove how the necessary condition will be applied throughout the entire interval of decrease of one or another descending position of the vector. In a confined space, we will select a variable from the initial block of our script. A system constructed as a basis along three vectors is responsible for the absence of the main moment of force. However, the equation calculator generated and helped in finding all the terms of the constructed equation, both above the surface and along parallel lines. Let's draw a circle around the starting point. Thus, we will begin to move up along the section lines, and the tangent will describe the circle along its entire length, resulting in a curve called an involute. By the way, let's tell a little history about this curve. The fact is that historically in mathematics there was no concept of mathematics itself in its pure understanding as it is today. Previously, all scientists were engaged in one common task, that is, science. Later, several centuries later, when the scientific world was filled with a colossal amount of information, humanity nevertheless identified many disciplines. They still remain unchanged. And yet, every year, scientists around the world try to prove that science is limitless, and you won't solve the equation unless you have knowledge of the natural sciences. It may not be possible to finally put an end to it. Thinking about this is as pointless as warming the air outside. Let us find the interval at which the argument, if its value is positive, will determine the modulus of the value in a sharply increasing direction. The reaction will help you find at least three solutions, but you will need to check them. Let's start with the fact that we need to solve the equation online using the unique service of our website. Let’s enter both sides of the given equation, click on the “SOLVE” button and get the exact answer within just a few seconds. In special cases, let's take a book on mathematics and double-check our answer, namely, look only at the answer and everything will become clear. The same project for an artificial redundant parallelepiped will fly out. There is a parallelogram with its parallel sides, and it explains many principles and approaches to studying the spatial relationship of the ascending process of accumulating hollow space in natural form formulas. Ambiguous linear equations show the dependence of the desired variable on our general solution at a given time, and we must somehow derive and bring the improper fraction to a nontrivial case. Mark ten points on the straight line and draw a curve through each point in the given direction, with the convex point up. Without any special difficulties, our equation calculator will present an expression in such a form that its check for the validity of the rules will be obvious even at the beginning of the recording. The system of special representations of stability for mathematicians comes first, unless otherwise provided by the formula. We will respond to this with a detailed presentation of a report on the topic of the isomorphic state of a plastic system of bodies and solving equations online will describe the movement of each material point in this system. At the level of in-depth research, it will be necessary to clarify in detail the issue of inversions of at least the lower layer of space. Ascending in the section where the function is discontinuous, we will apply the general method of an excellent researcher, by the way, our fellow countryman, and will tell below about the behavior of the plane. Due to the strong characteristics of an analytically defined function, we only use the online equation calculator for its intended purpose within the derived limits of authority. Reasoning further, we will focus our review on the homogeneity of the equation itself, that is, its right side is equal to zero. Let us once again make sure that our decision in mathematics is correct. In order to avoid obtaining a trivial solution, we will make some adjustments to the initial conditions for the problem of conditional stability of the system. Let's create a quadratic equation, for which we write out two entries using a well-known formula and find the negative roots. If one root is five units larger than the second and third roots, then by making changes to the main argument we thereby distort the initial conditions of the subtask. By its very nature, something unusual in mathematics can always be described to the nearest hundredth of a positive number. The fraction calculator is several times superior to its analogues on similar resources at the best moment of server load. On the surface of the velocity vector growing along the ordinate axis, we draw seven lines, bent in directions opposite to each other. The commensurability of the assigned function argument is ahead of the readings of the recovery balance counter. In mathematics, we can represent this phenomenon through a cubic equation with imaginary coefficients, as well as in the bipolar progression of decreasing lines. Critical points of temperature difference in many of their meaning and progression describe the process of decomposing a complex fractional function into factors. If you are told to solve an equation, do not rush to do it right away, definitely first evaluate the entire action plan, and only then take the right approach. There will certainly be benefits. The ease of work is obvious, and the same is true in mathematics. Solve the equation online. All online equations represent a certain type of record of numbers or parameters and a variable that needs to be determined. Calculate this very variable, that is, find specific values ​​or intervals of a set of values ​​at which the identity will hold. The initial and final conditions directly depend. The general solution of equations usually includes some variables and constants, by setting which we will obtain entire families of solutions for a given problem statement. In general, this justifies the efforts invested in increasing the functionality of a spatial cube with a side equal to 100 centimeters. You can apply a theorem or lemma at any stage of constructing an answer. The site gradually produces an equation calculator if it is necessary to show the smallest value at any interval of summation of products. In half the cases, such a ball, being hollow, no longer meets the requirements for setting an intermediate answer. At least on the ordinate axis in the direction of decreasing vector representation, this proportion will undoubtedly be more optimal than the previous expression. At the hour when a complete point analysis is carried out on linear functions, we will, in fact, bring together all our complex numbers and bipolar planar spaces. By substituting a variable into the resulting expression, you will solve the equation step by step and give the most detailed answer with high accuracy. It would be good form on the part of a student to check his actions in mathematics once again. The proportion in the ratio of fractions recorded the integrity of the result in all important areas of activity of the zero vector. Triviality is confirmed at the end of the completed actions. With a simple task, students may not have any difficulties if they solve the equation online in the shortest possible time, but do not forget about all the different rules. A set of subsets intersect in a region of convergent notation. In different cases, the product is not erroneously factorized. You will be helped to solve the equation online in our first section, dedicated to the basics of mathematical techniques for important sections for students in universities and technical colleges. We won’t have to wait a few days for answers, since the process of the best interaction of vector analysis with sequential finding of solutions was patented at the beginning of the last century. It turns out that the efforts to establish relationships with the surrounding team were not in vain; something else was obviously needed first. Several generations later, scientists all over the world made people believe that mathematics is the queen of sciences. Whether it is the left answer or the right one, all the same, the exhaustive terms must be written in three rows, since in our case we will definitely be talking only about vector analysis of the properties of the matrix. Nonlinear and linear equations, along with biquadratic equations, took a special place in our book about the best methods for calculating the trajectory of motion in the space of all material points of a closed system. A linear analysis of the scalar product of three consecutive vectors will help us bring the idea to life. At the end of each statement, the task is made easier by implementing optimized numerical exceptions across the number space overlays being performed. A different judgment will not contrast the found answer in the arbitrary shape of a triangle in a circle. The angle between two vectors contains the required percentage of margin, and solving equations online often reveals a certain common root of the equation as opposed to the initial conditions. The exception plays the role of a catalyst in the entire inevitable process of finding a positive solution in the field of defining a function. If it is not said that you cannot use a computer, then an online equation calculator is just right for your difficult problems. You just need to enter your conditional data in the correct format and our server will issue a full-fledged resulting answer in the shortest possible time. An exponential function increases much faster than a linear one. The Talmuds of smart library literature testify to this. Will perform a calculation in the general sense as a given quadratic equation with three complex coefficients would do. The parabola in the upper part of the half-plane characterizes rectilinear parallel motion along the axes of the point. Here it is worth mentioning the potential difference in the working space of the body. In return for a suboptimal result, our fraction calculator rightfully occupies the first position in the mathematical rating of the review of functional programs on the server side. The ease of use of this service will be appreciated by millions of Internet users. If you don’t know how to use it, we will be happy to help you. We would also like to especially note and highlight the cubic equation from a number of primary school problems, when it is necessary to quickly find its roots and construct a graph of the function on a plane. Higher degrees of reproduction are one of the complex mathematical problems at the institute and a sufficient number of hours are allocated for its study. Like all linear equations, ours are no exception according to many objective rules; look from different points of view, and it turns out to be simple and sufficient to set the initial conditions. The interval of increase coincides with the interval of convexity of the function. Solving equations online. The study of theory is based on online equations from numerous sections on the study of the main discipline. In the case of this approach in uncertain problems, it is very simple to present the solution to equations in a predetermined form and not only draw conclusions, but also predict the outcome of such a positive solution. A service in the best traditions of mathematics will help us learn the subject area, just as it is customary in the East. At the best moments of the time interval, similar tasks were multiplied by a common factor of ten. The abundance of multiplications of multiple variables in the equation calculator began to multiply by quality rather than quantitative variables such as mass or body weight. In order to avoid cases of imbalance of the material system, the derivation of a three-dimensional transformer on the trivial convergence of non-degenerate mathematical matrices is quite obvious to us. Complete the task and solve the equation in the given coordinates, since the conclusion is unknown in advance, as are all the variables included in post-space time. For a short time, move the common factor out of the parentheses and divide both sides by the greatest common factor in advance. From under the resulting covered subset of numbers, extract in a detailed manner thirty-three points in a row in a short period. To the extent that it is possible for every student to solve an equation online in the best possible way, looking ahead, let’s say one important but key thing, without which it will be difficult to live in the future. In the last century, the great scientist noticed a number of patterns in the theory of mathematics. In practice, the result was not quite the expected impression of the events. However, in principle, this very solution of equations online helps to improve the understanding and perception of a holistic approach to studying and practical consolidation of the theoretical material covered by students. It is much easier to do this during your study time.

=

The online fraction calculator allows you to perform simple arithmetic operations with fractions: adding fractions, subtracting fractions, multiplying fractions, dividing fractions. To make calculations, fill in the fields corresponding to the numerators and denominators of the two fractions.

Fractions in mathematics is a number representing a part of a unit or several parts of it.

A common fraction is written as two numbers, usually separated by a horizontal line indicating the division sign. The number above the line is called the numerator. The number below the line is called the denominator. The denominator of a fraction shows the number of equal parts into which the whole is divided, and the numerator of the fraction shows the number of these parts of the whole taken.

Fractions can be regular or improper.

  • A fraction whose numerator is less than its denominator is called a proper fraction.
  • An improper fraction is when the numerator of a fraction is greater than the denominator.

A mixed fraction is a fraction written as an integer and a proper fraction, and is understood as the sum of this number and the fractional part. Accordingly, a fraction that does not have an integer part is called a simple fraction. Any mixed fraction can be converted to an improper fraction.

In order to convert a mixed fraction into a common fraction, you need to add the product of the whole part and the denominator to the numerator of the fraction:

How to convert a common fraction to a mixed fraction

In order to convert an ordinary fraction to a mixed fraction, you must:

  1. Divide the numerator of a fraction by its denominator
  2. The result of division will be the whole part
  3. The balance of the department will be the numerator

How to convert a fraction to a decimal

In order to convert a fraction to a decimal, you need to divide its numerator by its denominator.

In order to convert a decimal fraction to an ordinary fraction, you must:


How to convert a fraction to a percentage

In order to convert a common or mixed fraction to a percentage, you need to convert it to a decimal fraction and multiply by 100.

How to convert percentages to fractions

In order to convert percentages into fractions, you need to obtain a decimal fraction from the percentage (dividing by 100), then convert the resulting decimal fraction into an ordinary fraction.

Adding Fractions

The algorithm for adding two fractions is as follows:

  1. Perform addition of fractions by adding their numerators.

Subtracting Fractions

Algorithm for subtracting two fractions:

  1. Convert mixed fractions to ordinary fractions (get rid of the whole part).
  2. Reduce fractions to a common denominator. To do this, you need to multiply the numerator and denominator of the first fraction by the denominator of the second fraction, and multiply the numerator and denominator of the second fraction by the denominator of the first fraction.
  3. Subtract one fraction from another by subtracting the numerator of the second fraction from the numerator of the first.
  4. Find the greatest common divisor (GCD) of the numerator and denominator and reduce the fraction by dividing the numerator and denominator by GCD.
  5. If the numerator of the final fraction is greater than the denominator, then select the whole part.

Multiplying fractions

Algorithm for multiplying two fractions:

  1. Convert mixed fractions to ordinary fractions (get rid of the whole part).
  2. Find the greatest common divisor (GCD) of the numerator and denominator and reduce the fraction by dividing the numerator and denominator by GCD.
  3. If the numerator of the final fraction is greater than the denominator, then select the whole part.

Division of fractions

Algorithm for dividing two fractions:

  1. Convert mixed fractions to ordinary fractions (get rid of the whole part).
  2. To divide fractions, you need to transform the second fraction by swapping its numerator and denominator, and then multiply the fractions.
  3. Multiply the numerator of the first fraction by the numerator of the second fraction and the denominator of the first fraction by the denominator of the second.
  4. Find the greatest common divisor (GCD) of the numerator and denominator and reduce the fraction by dividing the numerator and denominator by GCD.
  5. If the numerator of the final fraction is greater than the denominator, then select the whole part.

Online calculators and converters:

to solve mathematics. Find quickly solving a mathematical equation in mode online. The website www.site allows solve the equation almost any given algebraic, trigonometric or transcendental equation online. When studying almost any branch of mathematics at different stages you have to decide equations online. To get an answer immediately, and most importantly an accurate answer, you need a resource that allows you to do this. Thanks to the site www.site solve equations online will take a few minutes. The main advantage of www.site when solving mathematical equations online- this is the speed and accuracy of the response provided. The site is able to solve any algebraic equations online, trigonometric equations online, transcendental equations online, and equations with unknown parameters in mode online. Equations serve as a powerful mathematical apparatus solutions practical problems. With the help mathematical equations it is possible to express facts and relationships that may seem confusing and complex at first glance. Unknown quantities equations can be found by formulating the problem in mathematical language in the form equations And decide received task in mode online on the website www.site. Any algebraic equation, trigonometric equation or equations containing transcendental features you can easily decide online and get the exact answer. When studying natural sciences, you inevitably encounter the need solving equations. In this case, the answer must be accurate and must be obtained immediately in the mode online. Therefore for solving mathematical equations online we recommend the site www.site, which will become your indispensable calculator for solving algebraic equations online, trigonometric equations online, and transcendental equations online or equations with unknown parameters. For practical problems of finding the roots of various mathematical equations resource www.. Solving equations online yourself, it is useful to check the received answer using online equation solving on the website www.site. You need to write the equation correctly and instantly get online solution, after which all that remains is to compare the answer with your solution to the equation. Checking the answer will take no more than a minute, it’s enough solve equation online and compare the answers. This will help you avoid mistakes in decision and correct the answer in time when solving equations online either algebraic, trigonometric, transcendental or the equation with unknown parameters.


Let us analyze two types of solutions to systems of equations:

1. Solving the system using the substitution method.
2. Solving the system by term-by-term addition (subtraction) of the system equations.

In order to solve the system of equations by substitution method you need to follow a simple algorithm:
1. Express. From any equation we express one variable.
2. Substitute. We substitute the resulting value into another equation instead of the expressed variable.
3. Solve the resulting equation with one variable. We find a solution to the system.

To solve system by term-by-term addition (subtraction) method need to:
1. Select a variable for which we will make identical coefficients.
2. We add or subtract equations, resulting in an equation with one variable.
3. Solve the resulting linear equation. We find a solution to the system.

The solution to the system is the intersection points of the function graphs.

Let us consider in detail the solution of systems using examples.

Example #1:

Let's solve by substitution method

Solving a system of equations using the substitution method

2x+5y=1 (1 equation)
x-10y=3 (2nd equation)

1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, which means that it is easiest to express the variable x from the second equation.
x=3+10y

2.After we have expressed it, we substitute 3+10y into the first equation instead of the variable x.
2(3+10y)+5y=1

3. Solve the resulting equation with one variable.
2(3+10y)+5y=1 (open the brackets)
6+20y+5y=1
25y=1-6
25y=-5 |: (25)
y=-5:25
y=-0.2

The solution to the equation system is the intersection points of the graphs, therefore we need to find x and y, because the intersection point consists of x and y. Let's find x, in the first point where we expressed it we substitute y.
x=3+10y
x=3+10*(-0.2)=1

It is customary to write points in the first place we write the variable x, and in the second place the variable y.
Answer: (1; -0.2)

Example #2:

Let's solve using the term-by-term addition (subtraction) method.

Solving a system of equations using the addition method

3x-2y=1 (1 equation)
2x-3y=-10 (2nd equation)

1. We choose a variable, let’s say we choose x. In the first equation, the variable x has a coefficient of 3, in the second - 2. We need to make the coefficients the same, for this we have the right to multiply the equations or divide by any number. We multiply the first equation by 2, and the second by 3 and get a total coefficient of 6.

3x-2y=1 |*2
6x-4y=2

2x-3y=-10 |*3
6x-9y=-30

2. Subtract the second from the first equation to get rid of the variable x. Solve the linear equation.
__6x-4y=2

5y=32 | :5
y=6.4

3. Find x. We substitute the found y into any of the equations, let’s say into the first equation.
3x-2y=1
3x-2*6.4=1
3x-12.8=1
3x=1+12.8
3x=13.8 |:3
x=4.6

The intersection point will be x=4.6; y=6.4
Answer: (4.6; 6.4)

Do you want to prepare for exams for free? Tutor online for free. No kidding.