Regarding equation thinking, what is equation thinking

The idea of equation is the application of a problem to solve with equations, and it is also the understanding of the conceptual nature of equations, and the analysis of variables in mathematical problems.

construct equations or systems of equations, or use the properties of equations to analyze, transform, or solve problems. It is necessary to make good use of equations and systems of equations to observe and deal with problems. The idea of equations is to seek stillness in motion, and to study the equiquantity relationship in motion.

When a problem is likely to be related to an equation, the equation can be constructed and the properties of the equation can be studied to solve the problem. For example, when proving Cauchy's inequality, you can transform Cauchy's inequality into a discriminant of a quadratic equation.

The most basic thing is to be able to figure out the relationship between the meaning of the question and the quantity in it, use letters to represent the unknown, find a major equiquantitative relationship that can represent the meaning of the problem, and list the expressions according to the quantities involved in the relationship to obtain the equation.

When solving the problem, the equation is solved.

You need to think backwards, there are too many variables.

1. When you need to use equations.

2. When reverse thinking is not easy to solve, use forward thinking equations to solve it.

3. When there are 2 or more variables, the equation is more convenient to solve.

Equations have a variety of forms, such as unary linear equations, binary linear equations, unary quadratic equations, etc., and can also form a system of equations to solve multiple unknowns.

Basis for solving the equation.

1. Shift term change: move some terms in the equation from one side of the equation to the other side with the previous symbols, and add, subtract, subtract, multiply and divide, divide by multiplication;

2. The basic properties of the equation:

1) Add (or subtract) the same number or the same algebraic formula on both sides of the equation at the same time, and the result obtained is still the equation. It is expressed in letters as: if a=b, c is a number or an algebraic formula.

2) Multiply or divide both sides of the equation by a number that is not 0 at the same time, and the result is still the equation. It is expressed in letters as: if a=b, c is a number or an algebraic buried obelangue (not 0).

For example, for the calculation of (5+x)*6 2=48, the detailed steps are as follows:

5+x)*3=48,5+x=48/3,5+x=16,x=16-5,x=11.

That is, the solution of the equation sought.

5+x)*6/2=48,5+x)*3=48,5+x=48/3,5+x=16,x=16-5,x=11.

That is, the solution of the equation is returned.

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The calculation process for this problem is as follows:

Left = (5+x)*6 2=(5+11)*6 2=16*3=48, right = 48

Left = Right, i.e. loss x = 11 is the solution of the equation.

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Knowledge Expansion: A one-dimensional equation is an equation that contains only one unknown, the highest order of the unknown is 1, and both sides are integers. There is only one root of the unary equation, and the unary equation can solve most of the engineering problems, travel problems, distribution problems, profit and loss problems, integral table problems, billing problems, and number problems.

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Geometric Significance of Unary Linear Equations:

Since the unary linear functions can be converted into the form of ax+b=0 (a, b is constant, a≠0), the solution of the unary linear equation can be converted into the value of the corresponding independent variable when the value of a function is 0. From the image, this is equivalent to finding the abscissa of the intersection point of the line y=kx+b (k,b is a constant, k≠0) and the x-axis.

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The study of equations dates back to ancient times, when the ancient Egyptians wrote mathematical problems on papyrus more than 3,600 years ago that involved equations with hidden unknowns. Around 825 A.D., the mathematician Al-Khwarazmi, a mathematician from Central Asia, wrote a book entitled "Elimination and Reduction", focusing on the solution of equations, which had a great influence on the later development of mathematics.

For a long time, equations were not expressed in a specialized form, but were described in a general language. In the 17th century, the French mathematician Descartes first proposed to use letters such as xy and z to represent unknowns, and these letters were treated the same as ordinary numbers. Later, after continuous simplification and improvement, the equation gradually evolved into the current form of expression, such as 6x+8=20, 4x-2y=9, x-4=0, etc.

China also has a long history of studying equations. The Nine Chapters of Arithmetic, an ancient Chinese mathematical work, was written around 20050 BC and includes a chapter dedicated to "equations". The equations mentioned in this chapter are actually what people now call a system of equations, which is composed of several equations combined, and its solution is the common solution of these equations.

In the chapter "Equations", some practical problems are taken as examples, and methods of solving problems using systems of equations are given. >

When ancient Chinese mathematicians represented equations, they only used arithmetic chips to represent the coefficients of each unknown, and did not use special notation to represent unknowns. According to this notation, the system of equations is arranged into a rectangular square of numbers, which is very close to matrices in modern mathematics. When Liu Hui, an ancient Chinese mathematician, commented on the meaning of "equation", he pointed out that the word "square" is closely related to the above-mentioned number square, and the word "cheng" refers to the equation with unknown numbers, so the Chinese word "equation".

The term was first used to set out a set of equations with unknowns to solve a practical problem. During the Song and Yuan dynasties, Chinese mathematicians founded the "Tian Yuan Technique", using "Tian Jing Yuan to represent unknown numbers and establish equations, the representative work of this method is the mathematician Li Zhi wrote the "Circle Sea Mirror", the book said "Li Tian Yuan Yi" is equivalent to the current "set the unknown number x".

With the continuous expansion of the scope of mathematical research, equations are commonly used, its role is increasing, and the types of equations are constantly developing from simple to complex. However, no matter how the type changes, the equations of the form and form are all equations with unknowns, and they all express the equivalence relationship involving the unknown bright front number. The basic idea of solving equations is to gradually reduce the unknown to the form of known numbers according to the equiquantity relation, which is the essence of the equation.

Solution: Let this angle be x, which is derived from the title.

90-x=x／5

The solution is x=75

So 180-x=105

Answer: The complementary angle of this angle is 105 degrees.

The first known condition is a+ 3= (ab +1), the square of both sides gives a+3+2 (3a)=ab+1, and simplifies to ab=a+2 (3a)+2.

The second known condition is multiplied by 2ab to ab=a+2b, substituting the above equation to obtain the dust loss b= (3a)+1, and then substituting back to ab=a+2b to obtain a* (3a)=2 (3a)+2.

Substituting b = (3a)+1 into the algebraic formula found, 3ab-3b-b 2- (3a)=3a*(3a)+1)-3*(3a)+1)-(Paiwangshen(3a)+1) 2- (3a)=3a* (3a)+3a-3* (3a)-3-3a-1-2* (3a)- 3a)=6* (3a)+6+3a-3* (3a)-3-3a-1-2* (3a)- 3a)=2.

Let the triangle have a waist length of x

Depending on the title, the column is resistant to vertical chaos x+x 2-(5+x 2)=3 or 5+x 2-(x+x 2)=3

In the first case, x=8

The second case x=2

The second case is excluded based on the fact that the sum of the two sides of the triangle is greater than the third side.

So x=8 is the waist length of 8

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I wish the owner of the fiber transport to learn and make progress and be happy every day!