Rounding numbers after 5. Rounding natural numbers

To solve a system of linear equations with two variables using the addition method, you need to:

1) multiply the left and right parts of one or both equations by a certain number so that the coefficients for one of the variables in the equations become opposite numbers;

2) fold term by term received equations and find the value of one of the variables;

3) substitute the found value of one variable into one of these equations and find the value of the second variable.

If in a given system the coefficients for one variable are opposite numbers, then we will start solving the system immediately from point 2).

Examples. Solve a system of linear equations with two variables using the addition method.

Since the coefficients at y are opposite numbers (-1 and 1), we start the solution from point 2). We add the equations term by term and get the equation 8x = 24. Any equation of the original system can be written as the second equation of the system.

Find x and substitute its value into the 2nd equation.

We solve the 2nd equation: 9-y \u003d 14, hence y \u003d -5.

Let's do verification. Substitute the values ​​x = 3 and y = -5 into the original system of equations.

Note. The check can be done orally and not recorded if the check is not specified in the condition.

Answer: (3; -5).

If we multiply the 1st equation by (-2), then the coefficients for the variable x will become opposite numbers:

We add these equalities term by term.

We will obtain an equivalent system of equations, in which the 1st equation is the sum of two equations of the previous system, and the 2nd equation of the system we will write the 1st equation of the original system ( usually write the equation with smaller coefficients):

We find at from the 1st equation and the resulting value is substituted into the 2nd.

We solve the last equation of the system and get x = -2.

Answer: (-2; 1).

Let's make the coefficients for the variable at opposite numbers. To do this, we multiply all the terms of the 1st equation by 5, and all the terms of the 2nd equation by 2.

Substitute the value x=4 into the 2nd equation.

3 · 4 - 5y \u003d 27. Let's simplify: 12 - 5y \u003d 27, hence -5y \u003d 15, and y \u003d -3.

Answer: (4; -3).

To solve a system of linear equations with two variables using the substitution method, we proceed as follows:

1) we express one variable through another in one of the equations of the system (x through y or y through x);

2) we substitute the resulting expression into another equation of the system and obtain a linear equation with one variable;

3) we solve the resulting linear equation with one variable and find the value of this variable;

4) the found value of the variable is substituted into expression (1) for another variable and we find the value of this variable.

Examples. Solve a system of linear equations using the substitution method.

Express X through y from the 1st equation. We get: x \u003d 7 + y. We substitute the expression (7 + y) instead of X into the 2nd equation of the system.

We got the equation: 3 · (7+y)+2y=16. This is a one variable equation at. We solve it. Let's open the brackets: 21+3y+2y=16. Collecting terms with a variable at on the left side, and the free terms on the right. When transferring a term from one part of the equality to another, we change the sign of the term to the opposite.

We get: 3y + 2y \u003d 16-21. We give like terms in each part of the equality. 5y=-5. We divide both sides of the equality by the coefficient of the variable. y=-5:5; y=-1. Substitute this value at into the expression x=7+y and find X. We get: x=7-1; x=6. A pair of variable values ​​x=6 and y=-1 is the solution to this system.

Write down: (6; -1). Answer: (6; -1). It is convenient to write these arguments as shown below, i.e., systems of equations - on the left under each other. On the right - calculations, necessary explanations, verification of the solution, etc.

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Systems of equations are widely used in the economic industry in the mathematical modeling of various processes. For example, when solving problems of production management and planning, logistics routes (transport problem) or equipment placement.

Equation systems are used not only in the field of mathematics, but also in physics, chemistry and biology, when solving problems of finding the population size.

A system of linear equations is a term for two or more equations with several variables for which it is necessary to find a common solution. Such a sequence of numbers for which all equations become true equalities or prove that the sequence does not exist.

Linear Equation

Equations of the form ax+by=c are called linear. The designations x, y are the unknowns, the value of which must be found, b, a are the coefficients of the variables, c is the free term of the equation.
Solving the equation by plotting its graph will look like a straight line, all points of which are the solution of the polynomial.

Types of systems of linear equations

The simplest are examples of systems of linear equations with two variables X and Y.

F1(x, y) = 0 and F2(x, y) = 0, where F1,2 are functions and (x, y) are function variables.

Solve a system of equations - it means to find such values ​​(x, y) at which the system turns into a true equality or establish that suitable values x and y do not exist.

A pair of values ​​(x, y), written as point coordinates, is called a solution to a system of linear equations.

If the systems have one common solution or there is no solution, they are called equivalent.

Homogeneous systems of linear equations are systems whose right side is equal to zero. If the right part after the "equal" sign has a value or is expressed by a function, such a system is not homogeneous.

The number of variables can be much more than two, then we should talk about an example of a system of linear equations with three variables or more.

Faced with systems, schoolchildren assume that the number of equations must necessarily coincide with the number of unknowns, but this is not so. The number of equations in the system does not depend on the variables, there can be an arbitrarily large number of them.

Simple and complex methods for solving systems of equations

There is no general analytical way to solve such systems, all methods are based on numerical solutions. The school mathematics course describes in detail such methods as permutation, algebraic addition, substitution, as well as the graphical and matrix method, the solution by the Gauss method.

The main task in teaching methods of solving is to teach how to correctly analyze the system and find the optimal solution algorithm for each example. The main thing is not to memorize a system of rules and actions for each method, but to understand the principles of applying a particular method.

The solution of examples of systems of linear equations of the 7th grade of the general education school program is quite simple and is explained in great detail. In any textbook on mathematics, this section is given enough attention. The solution of examples of systems of linear equations by the method of Gauss and Cramer is studied in more detail in the first courses of higher educational institutions.

Solution of systems by the substitution method

The actions of the substitution method are aimed at expressing the value of one variable through the second. The expression is substituted into the remaining equation, then it is reduced to a single variable form. The action is repeated depending on the number of unknowns in the system

Let's give an example of a system of linear equations of the 7th class by the substitution method:

As can be seen from the example, the variable x was expressed through F(X) = 7 + Y. The resulting expression, substituted into the 2nd equation of the system in place of X, helped to obtain one variable Y in the 2nd equation. The solution of this example does not cause difficulties and allows you to get the Y value. Last step this is a test of the received values.

It is not always possible to solve an example of a system of linear equations by substitution. The equations can be complex and the expression of the variable in terms of the second unknown will be too cumbersome for further calculations. When there are more than 3 unknowns in the system, the substitution solution is also impractical.

Solution of an example of a system of linear inhomogeneous equations:

Solution using algebraic addition

When searching for a solution to systems by the addition method, term-by-term addition and multiplication of equations by various numbers. The ultimate goal of mathematical operations is an equation with one variable.

Applications of this method require practice and observation. It is not easy to solve a system of linear equations using the addition method with the number of variables 3 or more. Algebraic addition is useful when the equations contain fractions and decimal numbers.

Solution action algorithm:

1. Multiply both sides of the equation by some number. As a result arithmetic operation one of the coefficients of the variable must become equal to 1.
2. Add the resulting expression term by term and find one of the unknowns.
3. Substitute the resulting value into the 2nd equation of the system to find the remaining variable.

Solution method by introducing a new variable

A new variable can be introduced if the system needs to find a solution for no more than two equations, the number of unknowns should also be no more than two.

The method is used to simplify one of the equations by introducing a new variable. The new equation is solved with respect to the entered unknown, and the resulting value is used to determine the original variable.

It can be seen from the example that by introducing a new variable t, it was possible to reduce the 1st equation of the system to a standard square trinomial. You can solve a polynomial by finding the discriminant.

It is necessary to find the value of the discriminant using the well-known formula: D = b2 - 4*a*c, where D is the desired discriminant, b, a, c are the multipliers of the polynomial. In the given example, a=1, b=16, c=39, hence D=100. If the discriminant is greater than zero, then there are two solutions: t = -b±√D / 2*a, if the discriminant is less than zero, then there is only one solution: x= -b / 2*a.

The solution for the resulting systems is found by the addition method.

A visual method for solving systems

Suitable for systems with 3 equations. The method consists in plotting graphs of each equation included in the system on the coordinate axis. The coordinates of the points of intersection of the curves will be the general solution of the system.

The graphic method has a number of nuances. Consider several examples of solving systems of linear equations in a visual way.

As can be seen from the example, two points were constructed for each line, the values ​​of the variable x were chosen arbitrarily: 0 and 3. Based on the values ​​of x, the values ​​for y were found: 3 and 0. Points with coordinates (0, 3) and (3, 0) were marked on the graph and connected by a line.

The steps must be repeated for the second equation. The point of intersection of the lines is the solution of the system.

In the following example, it is required to find a graphical solution to the system of linear equations: 0.5x-y+2=0 and 0.5x-y-1=0.

As can be seen from the example, the system has no solution, because the graphs are parallel and do not intersect along their entire length.

The systems from Examples 2 and 3 are similar, but when constructed, it becomes obvious that their solutions are different. It should be remembered that it is not always possible to say whether the system has a solution or not, it is always necessary to build a graph.

Matrix and its varieties

Matrices are used to briefly write down a system of linear equations. A matrix is ​​a special type of table filled with numbers. n*m has n - rows and m - columns.

A matrix is ​​square when the number of columns and rows is equal. A matrix-vector is a single-column matrix with an infinitely possible number of rows. A matrix with units along one of the diagonals and other zero elements is called identity.

An inverse matrix is ​​such a matrix, when multiplied by which the original one turns into a unit one, such a matrix exists only for the original square one.

Rules for transforming a system of equations into a matrix

With regard to systems of equations, the coefficients and free members of the equations are written as numbers of the matrix, one equation is one row of the matrix.

A matrix row is called non-zero if at least one element of the row is not equal to zero. Therefore, if in any of the equations the number of variables differs, then it is necessary to enter zero in place of the missing unknown.

The columns of the matrix must strictly correspond to the variables. This means that the coefficients of the variable x can only be written in one column, for example the first, the coefficient of the unknown y - only in the second.

When multiplying a matrix, all matrix elements are sequentially multiplied by a number.

Options for finding the inverse matrix

The formula for finding the inverse matrix is ​​quite simple: K -1 = 1 / |K|, where K -1 is the inverse matrix and |K| - matrix determinant. |K| must not be equal to zero, then the system has a solution.

The determinant is easily calculated for a two-by-two matrix, it is only necessary to multiply the elements diagonally by each other. For the "three by three" option, there is a formula |K|=a 1 b 2 c 3 + a 1 b 3 c 2 + a 3 b 1 c 2 + a 2 b 3 c 1 + a 2 b 1 c 3 + a 3 b 2 c 1 . You can use the formula, or you can remember that you need to take one element from each row and each column so that the column and row numbers of the elements do not repeat in the product.

Solution of examples of systems of linear equations by the matrix method

The matrix method of finding a solution makes it possible to reduce cumbersome entries when solving systems with a large number of variables and equations.

In the example, a nm are the coefficients of the equations, the matrix is ​​a vector x n are the variables, and b n are the free terms.

Solution of systems by the Gauss method

In higher mathematics, the Gauss method is studied together with the Cramer method, and the process of finding a solution to systems is called the Gauss-Cramer method of solving. These methods are used to find the variables of systems with a large number of linear equations.

The Gaussian method is very similar to substitution and algebraic addition solutions, but is more systematic. In the school course, the Gaussian solution is used for systems of 3 and 4 equations. The purpose of the method is to bring the system to the form of an inverted trapezoid. By algebraic transformations and substitutions, the value of one variable is found in one of the equations of the system. The second equation is an expression with 2 unknowns, and 3 and 4 - with 3 and 4 variables, respectively.

After bringing the system to the described form, the further solution is reduced to the sequential substitution of known variables into the equations of the system.

In school textbooks for grade 7, an example of a Gaussian solution is described as follows:

As can be seen from the example, at step (3) two equations were obtained 3x 3 -2x 4 =11 and 3x 3 +2x 4 =7. The solution of any of the equations will allow you to find out one of the variables x n.

Theorem 5, which is mentioned in the text, says that if one of the equations of the system is replaced by an equivalent one, then the resulting system will also be equivalent to the original one.

The Gauss method is difficult for students to understand high school, but is one of the most interesting ways to develop the ingenuity of children enrolled in the advanced study program in math and physics classes.

For ease of recording calculations, it is customary to do the following:

Equation coefficients and free terms are written in the form of a matrix, where each row of the matrix corresponds to one of the equations of the system. separates the left side of the equation from the right side. Roman numerals denote the numbers of equations in the system.

First, they write down the matrix with which to work, then all the actions carried out with one of the rows. The resulting matrix is ​​written after the "arrow" sign and continue to perform the necessary algebraic operations until the result is achieved.

As a result, a matrix should be obtained in which one of the diagonals is 1, and all other coefficients are equal to zero, that is, the matrix is ​​reduced to a single form. We must not forget to make calculations with the numbers of both sides of the equation.

This notation is less cumbersome and allows you not to be distracted by listing numerous unknowns.

The free application of any method of solution will require care and a certain amount of experience. Not all methods are applied. Some ways of finding solutions are more preferable in a particular area of ​​human activity, while others exist for the purpose of learning.

Round numbers in Excel in several ways. Using cell format and using functions. These two methods should be distinguished as follows: the first is only for displaying values ​​or printing, and the second is also for calculations and calculations.

With the help of functions it is possible exact rounding, up or down, to a user-specified digit. And the values ​​obtained as a result of calculations can be used in other formulas and functions. At the same time, rounding using the cell format will not give the desired result, and the results of calculations with such values ​​will be erroneous. After all, the format of the cells, in fact, does not change the value, only its display method changes. In order to quickly and easily understand this and not make mistakes, we will give a few examples.

How to round a number by cell format

Let's enter the value 76.575 in cell A1. By right-clicking, we call the "Format Cells" menu. You can do the same with the number tool on home page Books. Or press the hot key combination CTRL+1.

Select the number format and set the number of decimal places to 0.

Rounding result:

You can assign the number of decimal places in the "monetary" format, "financial", "percentage".

As you can see, rounding occurs according to mathematical laws. The last digit to be stored is increased by one if it is followed by a digit greater than or equal to "5".

Peculiarity this option: the more digits after the decimal point we leave, the more accurate the result will be.

How to round a number correctly in Excel

Using the ROUND() function (rounds to the number of decimal places required by the user). To call the "Function Wizard" use the fx button. The desired function is in the "Math" category.

Arguments:

1. "Number" - a link to a cell with desired value(A1).
2. "Number of digits" - the number of decimal places to which the number will be rounded (0 - to round to an integer, 1 - one decimal place will be left, 2 - two, etc.).

Now let's round an integer (not a decimal). Let's use the ROUND function:

• the first argument of the function is a cell reference;
• the second argument - with the sign "-" (to tens - "-1", to hundreds - "-2", to round the number to thousands - "-3", etc.).

How to round a number in Excel to thousands?

An example of rounding a number to thousands:

Formula: =ROUND(A3,-3).

You can round not only the number, but also the value of the expression.

Suppose there is data on the price and quantity of goods. It is necessary to find the cost to the nearest ruble (round to the nearest whole number).

The first argument of the function is a numeric expression for finding the cost.

How to round up and down in Excel

To round up, use the ROUNDUP function.

We fill in the first argument according to the already familiar principle - a link to a cell with data.

The second argument: "0" - rounds the decimal fraction to the integer part, "1" - the function rounds, leaving one decimal place, etc.

Formula: =ROUNDUP(A1,0).

Result:

To round down in Excel, use the ROUNDDOWN function.

Formula example: =ROUNDDOWN(A1,1).

Result:

The ROUNDUP and ROUNDDOWN formulas are used to round expression values ​​(products, sums, differences, etc.).

How to round to whole number in Excel?

To round up to a whole number, use the ROUNDUP function. To round down to a whole number, use the ROUNDDOWN function. The "ROUND" function and the cell format also allow rounding to an integer by setting the number of digits to "0" (see above).

Excel also uses the "SELECT" function to round to a whole number. It simply discards the decimal places. Basically, there is no rounding. The formula cuts off the numbers to the designated digit.

Compare:

The second argument is "0" - the function cuts off to an integer; "1" - up to a tenth; "2" - up to a hundredth, etc.

A special Excel function that will return only an integer is INTEGER. It has a single argument - "Number". You can specify numerical value or cell reference.

The disadvantage of using the "INTEGER" function is that it only rounds down.

You can round up to a whole number in Excel using the ROUNDUP and ROUNDDOWN functions. Rounding occurs up or down to the nearest whole number.

An example of using functions:

The second argument is an indication of the digit to which rounding should occur (10 - to tens, 100 - to hundreds, etc.).

Rounding to the nearest even integer is performed by the "EVEN" function, to the nearest odd - "ODD".

An example of their use:

Why does Excel round large numbers?

If large numbers are entered into spreadsheet cells (for example, 78568435923100756), Excel automatically rounds them by default like this: 7.85684E+16 is a feature of the General cell format. To avoid such a display of large numbers, you need to change the format of the cell with the data a large number to "Numeric" (the fastest way is to press the hot key combination CTRL + SHIFT + 1). Then the cell value will be displayed like this: 78,568,435,923,100,756.00. If desired, the number of digits can be reduced: "Main" - "Number" - "Reduce bit depth".

Fractional numbers in Excel spreadsheets can be displayed to varying degrees. accuracy:

• most simple method - on the tab " home» press the buttons « Increase bit depth" or " Decrease bit depth»;
• click right click by cell, in the drop-down menu, select " Cell Format...”, then the tab “ Number", select the format" Numerical”, determine how many decimal places there will be after the decimal point (2 decimal places are suggested by default);
• click the cell, on the tab " home» choose « Numerical", or go to " Other number formats...” and configure there.

Here's what the fraction 0.129 looks like if you change the number of decimal places in the cell format:

Please note that A1,A2,A3 have the same meaning, only the form of representation changes. In further calculations, not the value visible on the screen will be used, but original. For a novice spreadsheet user, this can be a little confusing. To really change the value, you need to use special functions, there are several of them in Excel.

Rounding formula

One of the commonly used rounding functions is ROUND. It works according to standard mathematical rules. Select a cell, click the " Insert function”, category “ Mathematical", we find ROUND

We define the arguments, there are two of them - herself fraction and amount discharges. We click " OK' and see what happens.

For example, the expression =ROUND(0.129,1) will give a result of 0.1. The zero number of digits allows you to get rid of the fractional part. Choosing a negative number of digits allows you to round the integer part to tens, hundreds, and so on. For example, the expression =ROUND(5,129,-1) will give 10.

Round up or down

Excel provides other tools that allow you to work with decimals. One of them - ROUNDUP, gives the most close number, more modulo. For example, the expression =ROUNDUP(-10,2,0) will give -11. The number of digits here is 0, which means we get an integer value. nearest integer, greater in modulus, - just -11. Usage example:

ROUNDDOWN similar to the previous function, but returns the closest value that is smaller in absolute value. The difference in the work of the above means can be seen from examples:

We often use rounding in Everyday life. If the distance from home to school is 503 meters. We can say, by rounding up the value, that the distance from home to school is 500 meters. That is, we have brought the number 503 closer to the more easily perceived number 500. For example, a loaf of bread weighs 498 grams, then by rounding the result we can say that a loaf of bread weighs 500 grams.

rounding- this is the approximation of a number to a “lighter” number for human perception.

The result of rounding is approximate number. Rounding is indicated by the symbol ≈, such a symbol reads “approximately equal”.

You can write 503≈500 or 498≈500.

Such an entry is read as “five hundred three is approximately equal to five hundred” or “four hundred ninety-eight is approximately equal to five hundred”.

Let's take another example:

44 71≈4000 45 71≈5000

43 71≈4000 46 71≈5000

42 71≈4000 47 71≈5000

41 71≈4000 48 71≈5000

40 71≈4000 49 71≈5000

In this example, numbers have been rounded to the thousands place. If we look at the rounding pattern, we will see that in one case the numbers are rounded down, and in the other - up. After rounding, all other numbers after the thousands place were replaced by zeros.

Number rounding rules:

1) If the figure to be rounded is equal to 0, 1, 2, 3, 4, then the digit of the digit to which the rounding is going does not change, and the rest of the numbers are replaced by zeros.

2) If the figure to be rounded is equal to 5, 6, 7, 8, 9, then the digit of the digit up to which the rounding is going on becomes 1 more, and the remaining numbers are replaced by zeros.

For example:

1) Round to the tens place of 364.

The digit of tens in this example is the number 6. After the six is ​​the number 4. According to the rounding rule, the number 4 does not change the tens digit. We write zero instead of 4. We get:

36 4 ≈360

2) Round to the hundreds place of 4781.

The hundreds digit in this example is the number 7. After the seven is the number 8, which affects whether the hundreds digit changes or not. According to the rounding rule, the number 8 increases the hundreds place by 1, and the rest of the numbers are replaced by zeros. We get:

47 8 1≈48 00

3) Round to the thousands place of 215936.

The thousands place in this example is the number 5. After the five is the number 9, which affects whether the thousands place changes or not. According to the rounding rule, the number 9 increases the thousands place by 1, and the remaining numbers are replaced by zeros. We get:

215 9 36≈216 000

4) Round to the tens of thousands of 1,302,894.

The thousand digit in this example is the number 0. After zero, there is the number 2, which affects whether the tens of thousands digit changes or not. According to the rounding rule, the number 2 does not change the digit of tens of thousands, we replace this digit and all digits of the lower digits with zero. We get:

130 2 894≈130 0000

If a exact value number is unimportant, then the value of the number is rounded off and you can perform computational operations with approximate values. The result of the calculation is called estimation of the result of actions.

For example: 598⋅23≈600⋅20≈12000 is comparable to 598⋅23=13754

An estimate of the result of actions is used in order to quickly calculate the answer.

Examples for assignments on the topic rounding:

Example #1:
Determine to what digit rounding is done:
a) 3457987≈3500000 b) 4573426≈4573000 c) 16784≈17000
Let's remember what are the digits on the number 3457987.

7 - unit digit,

8 - tens place,

9 - hundreds place,

7 - thousands place,

5 - digit of tens of thousands,

4 - hundreds of thousands digit,
3 is the digit of millions.
Answer: a) 3 4 57 987≈3 5 00 000 digit of hundreds of thousands b) 4 573 426 ≈ 4 573 000 digit of thousands c) 16 7 841 ≈17 0 000 digit of tens of thousands.

Example #2:
Round the number to 5,999,994 places: a) tens b) hundreds c) millions.
Answer: a) 5,999,994 ≈5,999,990 b) 5,999,99 4≈6,000,000 6,000,000.