equation

[fāng chéng]

Mathematical terminology

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This entry is made by Compilation and application of scientific encyclopedia entries of "Science Popularization China" to examine.

Equation refers to the equation containing unknown number Of equation 。 It is an equation that represents the equal relationship between two mathematical expressions (such as two numbers, functions, quantities and operations). The value of the unknown number that makes the equation true is called "solution" or "root". The process of solving the equation is called "solving the equation".

The difficulty of reverse thinking can be avoided by solving the equation, and the equation containing the quantity to be solved can be directly listed in the positive direction. Equations have many forms, such as univariate linear equation, binary linear equation, univariate quadratic equation, etc., and can also form equations to solve multiple unknown number 。

In mathematics, an equation is a statement containing an equation of one or more variables. Solving an equation involves determining which values of variables make the equation hold. Variables are also called unknowns, and the value of unknowns that meet the equality is called the solution of the equation.^[1]

Chinese name: equation
Foreign name: equation
Definition: Equality with unknowns
Discipline: mathematics

application area: Mathematics, science, etc
Pinyin: fāng chéng
inventor: F Vieta
Form: Unary linear equation, unary quadratic equation, etc

catalog

one Mathematical terminology
▪ development
▪ name
▪ definition
two Relationship between equation and equation
three Basis for solving equation

thirteen The problem of chicken and rabbit in the same cage
▪ Solution formula
▪ Equation solution
fourteen differential equation
▪ Ordinary differential equation
▪ partial differential equation

Mathematical terminology

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development

As early as 3600 years ago, Ancient Egypt The mathematical problems written on the papyrus involve equations containing unknowns.^[2]

Around 825, Central Asia Mathematician of Al Hualazimi I once wrote a book called《 Cancellation and reduction 》The book focuses on the solution of equations.

name

The Chinese word "equation" comes from the ancient mathematical monograph《 Chapter Nine Arithmetic 》The eighth volume is called "Equation". "Fang" means juxtaposition, and "Cheng" means use Calculation Represents vertical.

The eighth (first) of the volume is: today, there are three strokes of the upper grain, two strokes of the middle grain, one stroke of the lower grain, and thirty-nine strokes; Two strokes of the upper grain, three strokes of the middle grain, one stroke of the lower grain, and thirty-four strokes; One stroke of the upper grain, two strokes of the middle grain, three strokes of the lower grain, and twenty-six strokes. Ask the upper, middle and lower levels of Heshi to grasp each geometry? (Nowadays, there are 3 bundles of superior millet, 2 bundles of medium millet, and 1 bundle of inferior millet, and 39 buckets of millet are produced; there are 2 bundles of superior millet, 3 bundles of medium millet, and 1 bundle of inferior millet, and 34 buckets of millet are produced; there are 1 bundle of superior millet, 2 bundles of medium millet, and 3 bundles of inferior millet, and 26 buckets of millet are produced. How many buckets of millet can be produced from 1 bundle of superior millet, 1 bundle of medium millet, and 1 bundle of inferior millet respectively?)

The answer was, "Up the grain, up to one, nine buckets, one quarter bucket, up to one, four buckets, one quarter bucket, down to one, two buckets, three quarters bucket.".

The formula says: put three strokes of the upper grain, two strokes of the middle grain, and one stroke of the lower grain, and make thirty-nine strokes on the right. The middle and left rows are on the right. Take the right row, go up to the grain, take the middle row, and divide it straight. Multiply it and then divide it directly. However, in the middle row, those who can't get enough grain will ride on the left row and be divided into straight rows. On the left side, those who can't get enough of the grain down, the upper part is the law, and the lower part is the reality. The truth is the truth of lowering the grain. Find the middle grain, multiply the middle row by the law, and remove the grain. The rest is like Zhonghe's counting, which is the truth of Zhonghe. To get the upper grain, multiply the right row by the lower grain, and remove the lower grain and the middle grain. The number of the remaining grains is equal to that of the above grains, which is the truth of the grains. If everything is as the law, each will get a fight.

The above is a ternary linear equation set from "Nine Chapters of Arithmetic", and shows how to solve this equation set by eliminating the element with "multiplication and direct division".

Big Mathematicians in the Wei and Jin Dynasties Liu Hui Around 263 AD, he made a lot of notes for Nine Chapters of Arithmetic, and introduced the equation system: two things are repeated, three things are three, all like things. It is called equation because it is parallel to lines. He also created a more simple "mutual multiplication and cancellation" method than "direct division by multiplication" to solve the equations.

definition

Equation is an equation with unknown numbers, which is in the textbook of primary school Logical definition , and the equation with unknown number is not necessarily an equation strictly, such as 0x=0. The equation is strictly defined as follows:

In the form of

Equation of, where

and

Are two analytic expressions studied within the intersection of the domain, and at least one is not constant 。

Relationship between equation and equation

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The equation must be an equation, but the equation is not necessarily an equation.

Example: a+b=13 conforms to the equation, and there are unknowns. This is an equation, also an equation.

1+1=2 ，100×100=10000。 These two formulas agree equation But there are no unknowns, so they are not equations.

In the definition, the equation must be an equation, but there can be other equations, such as the above 1+1=2100 × 100=10000, which are all equations. Obviously, the range of equations is a little larger.

Basis for solving equation

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one . transposition Change sign : Move some items in the equation from one side of the equation to the other with the preceding symbol, and add, change, subtract, add, multiply, divide, and divide by;

two Basic properties of equality

Nature 1

Add (or subtract) the same number or the same on both sides of the equation Algebraic expression The result is still an equation. If a=b, c is a number or an algebraic expression. Then: (1)

(2)

Nature 2

When both sides of an equation are multiplied or divided by the same number that is not zero, the result is still an equation.

It is expressed in letters: if a=b, c is a number or an algebraic expression (not 0). Then:

A × c=b × c or

Nature 3

If a=b, then b=a (of equation Symmetry ）。

Nature 4

If a=b, b=c, then a=c (of equation Transitivity ）。

Equation solving steps

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Method 1: 1. Calculate first if you can; two conversion ——Calculation - Results

Method 2: Calculate from the beginning to the end when there is only one number left.

Related concepts

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equation Or short for equation , is containing unknown number Equation for. Namely: 1. The equation must contain One or one The algebraic expression of the above unknowns; two equation Is an equation, but an equation is not necessarily an equation.

unknown number: Generally, x.y.z is set as the unknown number, and others can also be set letter , all lowercase letters are OK.

“ second ” : The concept of degree in equation and Integral form The concept of "sub" is similar. It refers to the item with the highest number of unknowns among the items containing unknowns. The term with the highest degree is the degree of the equation.

Solution : The solution of the equation means that the root of the equation is the value of the equal unknowns on both sides of the equation. It refers to the solution of the equation of one variable, and the two can generally be used.

solve equations The process of finding the solution of the equation can also be said to be the process of finding the value of the unknown number in the equation, or the process of explaining that the equation has no solution is called solving the equation.

In the equation, Identity be called Identity equation ， Paradoxical be called Contradictory equation 。 When the unknown number is equal to a specific value Equal sign The equal value of both sides is called Conditional equation , for example, x+3=8, when x=5, the equal sign holds. To make the left and right sides of an equation equal unknown number The value of is called Solution of the equation 。

Same solution equation:

If the solutions of two equations are the same, then these two equations are called the same solution equations.

The same solution principle of the equation:

1. The equation obtained by adding or subtracting the same number or the same equation from both sides of the equation and the original equation are Isometric equation 。

2. The equation obtained by multiplying or dividing both sides of the equation by the same number that is not zero is the same solution equation with the original equation.

Integral equation : of equation both sides Both On integral form of unknown number The equation of is called Integral form Equation.

Fractional equation ： The denominator contains an unknown number The equation of is called fraction Equation.

Unary linear equation

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It contains only one unknown number and the number of unknowns is one Integral equation call Unary linear equation （linear equation with one unknown）。 The usual form is Ax+b=0 (a, b are constants, and a ≠ 0) 。

General solution

one
Denominator removal Multiply the least common multiple of each denominator on both sides of the equation.
two
Bracket removal Generally, remove the parentheses first, then the brackets, and finally the braces. But sometimes the order can be determined according to the situation to make the calculation simple. Can be based on distributive law 。
three
transposition Move the unknown terms in the equation to the other side of the equation, and the rest to the other side of the equation. Don't forget to change the sign when you move the terms. (For example, 5x=4x+8 is obtained from 5x=4x+8; unknowns are moved together)
four
Merge Similar Items The original equation is reduced to the form ax=b (a ≠ 0).
five
turn coefficient For one Both sides of the equation are divided by the coefficient of the unknown at the same time.
six
The solution of the equation is obtained.

For example:

3x=5×6

Solution: 3x=30

x=30÷3

x=10

(Note: it is better to align the equal sign when solving the equation)

instructional design

Teaching objectives

one
To enable students to preliminarily master the methods and steps of solving simple word problems with one variable linear equation; And will list simple word problems for solving the unary linear equation
two
Cultivate students' observation ability and improve their ability to analyze and solve problems
three
Make students form the good habit of thinking correctly

Key and difficult points

The method and steps of solving simple word problems with one variable linear equation

teaching process

1、 Ask questions from students' original cognitive structure

In elementary school arithmetic, we learned about using arithmetic to solve practical problems. Then, can a practical problem be solved by using a univariate linear equation? If so, how? What are the advantages of solving word problems with one variable linear equation compared with solving word problems with arithmetic method?

To answer these questions, let's look at the following example

Example 1: 3 times of a number minus 2 equals to the sum of a number and 4. Find a number

(First, use arithmetic method to solve the problem. Let the students answer and the teacher write on the blackboard.)

Solution 1: (4+2) ÷ (3-1)=3

Answer: A certain number is 3

(Next, solve it by algebraic method, with teacher's guidance and students' dictation)

Solution 2: If a number is x, then 3x-2=x+4

3x-2=x+4

Solution: (3-1) x=2+4

2x=2+4

2x=6

x=6÷2

x=3

When solved, x=3

Answer: A certain number is 3

Looking at the two solutions in Example 1, it is obvious that the arithmetic method is not easy to think, but the method of setting unknown numbers, listing equations and solving equations to obtain the solution of word problems has a sense of turning difficulties into easy ones, which is one of the purposes of learning how to solve word problems by using unitary linear equations

We know that the equation is an equation with unknown numbers, and the equation represents an equality relationship. Therefore, for any condition provided in a word problem, we should first find an equality relationship from it, and then express the equality relationship as an equation

In this lesson, we will use examples to illustrate how to find an equal relationship and how to transform the equal relationship into an equation

2、 Teachers and students jointly analyze and study the methods and steps of solving simple word problems with unitary linear equations

Example 2 After 15% of the flour stored in a flour warehouse is shipped out, there are still 42500 kilograms left. How much flour is there in this warehouse?

Common analysis of teachers and students:

one
What are the known quantity and unknown quantity given in this question?
two
What is the equal relationship between known quantity and unknown quantity? (Original weight - delivery weight=remaining weight)
three
If the original flour has x kg, how many kg of flour can be transported? How to arrange the equation by using the above equality relationship?

The above analysis process can be listed as follows:

Solution: If there were x kilograms of flour, then 15% x kilograms were shipped out. According to the question, x-15% x=42500,

x-15%x=42500

Solution: (1-15%) x=42500

85%x=42500

x=42500÷85%

x=50000

So x=50000

Answer: There were 50000 kilograms of flour

At this time, let the students discuss: Is there any other form of expression for the equality relation of this question besides the above form of expression? If yes, what is it?

(Also, original weight=delivered weight+remaining weight; original weight - remaining weight=delivered weight)

Teachers should point out that: (1) The expression form of these two equality relations is different from "original weight - shipped weight=remaining weight", but the essence is the same. You can choose one of the equality relations to formulate the equation

(2) The process of solving the equation in Example 2 is relatively simple, and students should pay attention to imitation

According to the analysis and solution process of Example 2, first of all, please think about the method and steps of solving word problems by univariate linear equations; Then, take the way of questioning to give feedback; Finally, according to the students' summary, the teacher summarizes as follows:

(1) Carefully examine the question and thoroughly understand the meaning of the question Understand known quantity, unknown quantity and their relationship ； Use the letter (such as x) to indicate the unknown number in the question

(2) According to the meaning of the title Find the equality relationship . (This is the key step)

(3) According to the relationship of equality, List the equation correctly That is, the equations listed should meet the requirement that the quantities on both sides should be equal; On both sides of the equation Algebraic expression The units of must be the same; The conditions in the question shall be fully utilized, and one condition shall not be missed or reused

(4) Find the solution of the listed equation

(5) test Clearly and completely Write the answer The test required here should be that the solution obtained by the test can not only make the equation hold, but also make the word problem meaningful

linear equation in two unknowns

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People's Education Edition The fourth chapter of Grade 7 Mathematics Volume II will be learned, and the ninth chapter of Grade 7 Mathematics Volume II will be learned. Speaking Einstein in English in the ninth grade of PEP will also involve

Definition of binary linear equation: one contains two unknowns and the degree of the unknowns is 1 Integral equation , called linear equation in two unknowns (linear equation of two unknowns)。
Binary linear equations Definition: an equation group consisting of two binary linear equations, called Binary linear equations (system of linear equation of two unknowns)。
The solution of the bivariate linear equation: the value of two unknowns that make the values on both sides of the bivariate linear equation equal is called the solution of the bivariate linear equation.
Solutions of binary linear equations: the two common solutions of binary linear equations are called solutions of binary linear equations.

General solution

Elimination: reduce the number of unknowns in the equation set from more to less, and solve them one by one.

There are two methods of elimination:

Substitution elimination

Example: solve the equation set x+y=5 ① 6x+13y=89 ②

Solution: from ① x=5-y ③ to ②, 6 (5-y)+13y=89, y=59/7

Take y=59/7 into ③ and get x=5-59/7, that is, x=- 24/7

∴ x=-24/7，y=59/7

This solution is Substitution Elimination method.

Addition, subtraction and elimination

Example: solve the equation set x+y=9 ① x-y=5 ②

Solution: ①+②, 2x=14, that is, x=7

Take x=7 into ①, and get 7+y=9, and get y=2

∴ x=7，y=2

This solution is Addition and subtraction Elimination method.

Binary linear equations Solution of There are three situations ：

1. There is a set of solutions

For example, the solution of the equation set x+y=5 ① 6x+13y=89 ② is x=- 24/7, y=59/7.

2. There are countless solutions

For example, the equation group x+y=6 ① 2x+2y=12 ②, because these two equations are actually one equation (also called "equation has two equal Real root ”）Therefore, there are countless sets of solutions to this kind of equations.

3. No solution

For example, equation group x+y=4 ① 2x+2y=10 ②, because equation ② is simplified to x+y=5, which is inconsistent with equation ①, so there is no solution to such equation group.

Unary quadratic equation

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The integral equation with an unknown number and the highest degree of the unknown number is 2 is called the quadratic equation in one unknown.

It is a qualitative transformation from the first equation to the second equation. Generally, the second equation is much more complex than the first equation both in concept and solution.

General form

(a≠0)

General solution

There are four general solutions:

1. Formula method (direct Square root France)

⒉ Matching method

three Factorization method

four cross multiplication

The cross multiplication can convert some quadratic Trinomial Factorize. The key to this method is to Coefficient of quadratic term A is broken into two factor a one ,a two Product a of one ·a two ，把 Constant term C is decomposed into two factors c one ,c two Product c of one ·c two , and make a one c two +a two c one If it is just a primary term b, it can be directly written into the result: Factor We should pay attention to observation, try, and realize that its essence is binomial multiplication The reverse process of. When the first coefficient is not 1, multiple tests are often required, and the symbol of each coefficient must be noted.

Example 1 Factorize 2x ² - 7x+3.

Analysis: first decompose the coefficient of quadratic term, write it in the upper left corner and lower left corner of the crosshair, then decompose the constant term

Don't write it on the upper right corner and lower right corner of the cross line, then cross multiply it and find the algebraic sum so that it is equal to the coefficient of the primary term

Decomposition quadratic term coefficient (only take Positive factor )：

2=1×2=2×1；

Decomposition constant term:

3=1×3=3×1=(-3)×(-1)=(-1)×(-3).

The following four situations are represented by drawing cross lines:

After observation, the fourth case is correct because Cross multiplication After, the sum of two algebras is exactly equal to one time Term coefficient －7.

The solution is: 2x ² - 7x+3=(x-3) (2x-1).

Generally, for the quadratic trinomial ax ²+bx+c (a ≠ 0), if the quadratic coefficient a can be decomposed into the product of two factors, that is, a=a one a two , the constant term c can be decomposed into the product of two factors, that is, c=c one c two , put a one ，a two ，c one ，c two , arranged as follows:

Multiply by diagonal intersection, and then add to get, if it is exactly equal to the coefficient b of the first term of the quadratic trinomial ax ²+bx+c, that is, a one c two +a two c one =b. Then the quadratic trinomial can be decomposed into two factors a one x+c one And a two x+c two The product is:

ax²+bx+c=(a one x+c one )(a two x+c two )。

Like this Decomposition coefficient by drawing cross lines To help us Quadratic trinomial The method of factoring is usually called cross multiplication 。

Summary:

① Factorization of the formula of x ²+(p+q) x+pq type

The characteristics of this kind of quadratic trinomial are: the coefficient of the quadratic term is 1; The constant term is the product of two numbers; The coefficient of the primary term is the sum of the two factors of the constant term. Therefore, the quadratic trinomial whose coefficient of some quadratic terms is 1 can be factorized directly: x ²+(p+q) x+pq=(x+p) (x+q)

② Factorization of kx ²+mx+n

If it can be decomposed into k=ac, n=bd, and ad+bc=m, then

kx²+mx+n=(ax+b)(cx+d)

1． Direct leveling method:

The direct square root method is a method to solve the quadratic equation with one variable. use Direct opening method Solution such as (x-m) ²=n (n ≥ 0)

Equation, whose solution is

2． Matching method ： Solving the equation ax ²+bx+c=0 (a ≠ 0) by the collocation method

First move the constant c to the right of the equation: ax ²+bx=- c

The coefficient of quadratic term is reduced to 1: x ²+(b/a) x=- c/a

Add the square of half of the coefficient of the primary term on both sides of the equation: x ²+(b/a) x+(b/2a) ²=- c/a+(b/2a) ²

The left side of the equation becomes a Perfect square On the right, a constant is calculated: (x+b/2a) ²=- c/a+(b/2a) ²

Last use Direct opening method solve

3． Formula method : Turn the quadratic equation of one variable into a general form, and then calculate Discriminant △=the value of b ² - 4ac, when b ² - 4ac<0, there is no solution in the real number range; When b ² - 4ac ≥ 0, the values of coefficients a, b, c are substituted into the root formula

Then we can get the root of the equation.

4． Factorization : Transforming the equation to one side is Fatal Frame , put the other side Quadratic trinomial It is decomposed into the form of the product of two linear factors, so that

Two linear factors are equal to zero respectively , two unary linear equations are obtained, and the roots obtained by solving these two unary linear equations are the two roots of the original equation. This solution Unary quadratic equation The method of Factorization 。

Binary quadratic equation : with two unknowns and the maximum number of unknowns is 2 Integral form Equation.

Ternary linear equation

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Ternary linear equation

Similar to the binary linear equation, three linear equations with three unknowns are combined.

solution

Similar to the binary linear equation, the elimination method can be used to eliminate the element step by step.

Analysis of typical problems

In order to encourage water conservation, a certain area has made the following provisions on the charging standard of tap water: 0.9 yuan/ton will be charged if the water consumption per household does not exceed 10 tons per month; 1.6 yuan/ton for more than 10 tons but not more than 20 tons; The part exceeding 20 tons will be charged at 2.4 yuan/ton. In one month, user A paid 16 yuan more than user B, and user B paid 7.5 yuan more than user C. It is known that user C uses less than 10 tons of water and user B uses more than 10 tons but less than 20 tons of water. Q: A. B. How much is the monthly water fee paid by the third and third users (calculated by the whole ton)?

Solution: Assume that Party A uses x tons of water, Party B uses y tons of water, and Party C uses z tons of water

Obviously, user A uses more than 20 tons of water

Therefore, Party A's payment: 0.9 * 10+1.6 * 10+2.4 * (x-20)=2.4x-23

Payment by Party B: 0.9 * 10+1.6 * (y-10)=1.6y-9

Payment by Party C: 0.9z

2.4x-23=1.6y-7+16

1.6y-7=0.9z+7.5

Simplified

3x-2y=40……(1)

16y-9z=145……(2)

From (1), x=(2y+40)/3

So let y=1+3k, 3 ＜ k ＜ 7

When k=4, y=13, x=22, substitute (2) to get z=7

When k=5, y=16, substitute (2), z is not integer solution

When k=6, y=19, substitute into (2), z has no integer solution

Therefore, 22 tons of water is used for Party A, 13 tons for Party B and 7 tons for Party C

Water for Party A is 29.8 yuan, water for Party B is 13.8 yuan, and water for Party C is 6.3 yuan

Multivariate linear equation

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Elimination method

set up Equations ①：

……………………

Put equation (1) × (- i one /a one ）Add to (i), and then put equation (2) × (- i two /a two ）Add to (i), and so on. (i ∈ N and i ∈ [1, m])

The last many 0=0 can be omitted without affecting the solution of the equation. There are three situations:

（1）c r +1 ≠0

At this time, any solution that satisfies the previous r equations cannot satisfy 0=c r +1 This equation, so ② has no solution, so ① also has no solution

When c r +When 1=0, there are two cases:

（2）r=n

Because b ii ≠ 0, so x can be solved from the last equation n 。 then Substitution The r-1 equation, solve x n -1。 By analogy, the unique solution of equation set ② is the unique solution of equation set ①.

（3）r＜n

The equation set can be transformed into his equation set with the same solution ③:

b eleven x one +b twelve x two +b thirteen x three +…+b 1r x r =c one -b one ,r+1 x r +1-…-b 1n x n

b twenty-two x two +b thirteen x three +…+b 2n x r =c two -b two ,r+1 x r +1-…-b 2n x n

………………

b rr x r =c r -b r ,r+1 x r +1-…-b rn x n

Let the number following the equal sign be a known number, and solve it according to the method in (2), we can get:

x one =d eleven x r +1+d twelve x r +2+…+d one ,n-r x n

x two =d twenty-one x r +1+d twenty-two x r +2+…+d one ,n-r x n

………………

x r =d r1 x r +1+d r2 x r +2+…+d r ,n-r x n

Let the free unknown xr+i=ki (i ∈ N and i ∈ [1, n-r]) obtain all the solutions of the equation system:

x one =d eleven k one +d twelve k two +…+ d one ,n-r k n-r

x two =d twenty-one k one +d twenty-two k two +…+d one ,n-r k n-r

………………

x r =d r1 k one +d r2 k two +…+d r ,n-r k n-r

x r +1=k one

x r +2=k two

…………

x n =k n-r

Linear equation

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(1) General formula: Ax+By+C=0 (where A and B are different and 0) applies to all straight lines

Linear equation

Straight line l 1： A one x+B one y+C one =0

Straight line l 2： A two x+B two y+C two =0

When two straight lines are parallel: A1/A2=B1/B2 ≠ C1/C2

When two straight lines are vertical: A1A2+B1B2=0

When two straight lines coincide: A1/A2=B1/B2=C1/C2

When two straight lines intersect: A1/A2 ≠ B1/B2

(2) Point oblique type : Know a point on the line (x zero ,y zero ）, and The slope of the line If k exists, the line can be expressed as y-y zero =k(x-x zero )。 When k does not exist, the line can be expressed as x=x zero

(3) Intercept : If the line intersects the x axis at (a, 0) and the y axis at (0, b), the line can be expressed as: x/a+y/b=1. Therefore, it is not applicable to lines perpendicular to any coordinate axis and lines passing through the origin.

(4) Oblique cut type ： y=kx+b (k≠0)

(5) Two-point formula : If the straight line passes through any two points (x one ,y one )、(x two ,y two ）, and x one ≠x two ,y one ≠y two , the line can be expressed as

(6) Normal ： x·cosα+ysinα-p=0

note appended

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Generally, n-ary linear equation is an equation with n unknowns and the degree of the unknowns is 1 coefficient It is not equal to 0

N-ary linear equations are equations composed of several n-ary linear equations（ Unary linear equation Except)

An equation of degree a with one variable is an equation that contains an unknown number and the highest degree of the item containing the unknown number is a (except the equation of degree a with one variable)

A system of equations of degree a in one variable is a system of equations composed of several equations of degree a in one variable (except for the equations of degree a in one variable)

The n-variable a-degree equation is an equation containing n unknowns, and the highest degree of the unknowns is a (except the unary linear equation)

The n-ary a-degree equation system is the equation system composed of several n-ary a-degree equations (except the univariate first-degree equation)

The equation (group) in which the number of unknowns is greater than the number of equations is called Indefinite equation (group), such equations (groups) generally have countless solutions.

The problem of chicken and rabbit in the same cage

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Solution formula

Solution 1: (number of rabbit feet × total number of feet - total number of feet) ÷ (number of rabbit feet - number of chicken feet)=number of chickens

Total number - number of chickens=number of rabbits

Solution 2: (total number of feet - number of chicken feet × total number of feet) ÷ (number of rabbit feet - number of chicken feet)=number of rabbits

Total number - number of rabbits=number of chickens

Solution 3: total number of feet ÷ 2 - total number of heads=number of rabbits

Total number - number of rabbits=number of chickens

Solution 4 (equation): X=total number of feet ÷ 2 - total number of heads (X=number of rabbits)

Total number - number of rabbits=number of chickens

Solution 5 (equation): X=(total number of feet - number of chicken feet × total number of feet) ÷ (number of rabbit feet - number of chicken feet) (X=number of rabbits)

Total number - number of rabbits=number of chickens

Solution 6 (equation): X=(number of rabbit feet × total number of feet - total number of feet) ÷ (number of rabbit feet - number of chicken feet) (X=number of chicken feet)

Total number - number of chickens=number of rabbits

Equation solution

If equation solution is used The problem of chicken and rabbit in the same cage , the formula is: chicken feet+rabbit feet=total feet.

Chicken is x

There are 30 chickens and rabbits in the cage, and the number is exactly 100. How many chickens and rabbits are there?

Solution: If there are x chickens, there are (30-x) rabbits.

2x+(30-x)×4=100

Solution: 2x+120-4x=100

120-2x=100

2x=20

x=10

30-10=20 (pieces)

Answer: There are 10 chickens and 20 rabbits.

Rabbit is x

There are 100 chickens and rabbits in the cage, and 248 chickens and rabbits in total. How many chickens and rabbits are there?

Solution: If there are x rabbits, then there are (100-x) chickens.

4x+(100-x)×2=248

Solution: 4x+200-2x=248

2x+200=248

2x=48

x=24

100-24=76 (pieces)

Answer: There are 76 chickens and 24 rabbits.

differential equation

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differential equation Describing an unknown function derivatives And independent variable The relationship between the equations. The solution of the differential equation is a function conforming to the equation. In elementary mathematics, the solution of algebraic equation is constant. See differential equation

Differential equation is a mathematical equation that relates some functions with their derivatives. In applications, functions usually represent physical quantities, derivatives represent their rate of change, and equations define the relationship between them. Because this relationship is very common, differential equations play a prominent role in many disciplines including engineering, physics, economics and biology.

In pure mathematics, differential equations are studied from several different perspectives, mainly involving their solutions - function sets satisfying equations. Only the simplest differential equation can be solved by explicit formula; However, some properties of the solution of a given differential equation can be determined without finding its exact form.

If the solution's self-contained formula is not available, you can use computer values to approximate the solution. Dynamic system theory emphasizes the qualitative analysis of systems described by differential equations, and many numerical methods have been developed to determine solutions with a given accuracy.^[3]

Ordinary differential equation

Ordinary differential equation or ODE is an equation containing a function of an independent variable and its derivative. In contrast to "partial differential equations", the term "ordinary" relates to more than one independent variable.

Having solutions that can be added and multiplied by coefficients linear differential equation It is clearly defined and understood, and the exact closed form solution is obtained. In contrast, ODE, which lacks additive solutions, is nonlinear, and it is very complex to solve them, because they are rarely represented by basic functions in closed form: on the contrary, the precision and analytical solutions of ODE are in series or in the whole form. The graphical and numerical methods applied manually or by computer can approximate the ODE solution, and may generate useful information, which is usually sufficient in the absence of an accurate analytical solution.

partial differential equation

Partial differential equation (PDE) is a kind of differential equation containing unknown multivariable function and its partial derivative. (This is the opposite of ordinary differential equations that deal with functions of a single variable and its derivatives). PDE is used to formulate problems involving functions of several variables, or solve them manually or create relevant computer models.

PDE can be used to describe various phenomena, such as sound, heat, static electricity, electrodynamics, fluid flow, elasticity or quantum mechanics. These seemingly different physical phenomena can be similar to topographic formalization in PDE. Just as ordinary differential equations often simulate one-dimensional dynamic systems, partial differential equations usually simulate multidimensional systems. PDEs in Stochastic partial differential equation Find their generalization in.^[4]

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