What is the difference between the substitution method and the addition and subtraction method of el

The substitution method is to bring an unknown number into another analytic formula to operate, and it is to convert 1,2 formulas to the same unknown number, and then calculate.

The addition, subtraction, and elimination method is to directly add and subtract an equation to remove an unknown number, turn it into an equation containing only one unknown, and then calculate.

1. Definition and steps of alternative elimination method.

1 Alternative elimination method.

Place a binary linear equation.

The unknowns of one equation in the group are expressed as an equation with another unknown, which is then substituted by another equation to eliminate to obtain the solution of the system of binary linear equations. This method is called the alternative elimination method, or substitution method for short.

2. The general procedure for solving a system of binary linear equations by the substitution elimination method.

1) Transform one of the equations so that one unknown is represented by an algebraic expression that contains another unknown.

Denote; 2) substituting this algebraic expression for the corresponding unknowns in another equation to obtain a univariate linear equation;

3) solve a primary equation with a variable;

4) Substituting the value of an unknown quantity into any one of the algebraic formulas or the original equation to obtain the value of another unknown quantity;

5) Write the solution of the equation.

Three. Considerations for alternative elimination methods.

1) When using the substitution method to eliminate elements, the relationship derived from one equation in the system of equations must be substituted into another equation. If you substitute the original equation, it is impossible to find the solution of the original system of equations.

2) When the coefficients in the equation are not all integers, they should be simplified first, that is, they should be reduced to integer coefficients by using the properties of the equation.

3) When an unknown is found, it is easy to find its value by substituting another unknown into the transformed equation $y=$$ax+$$b$ (or $x=$$ay+b$).

4) In order to test whether the obtained pair of values is the solution of the original system of equations, you can substitute this pair of values into each equation of the original system of equations. If all equations are true, then this pair of values is the solution of the original system of equations, otherwise it means that the solution is wrong.

2. Examples of alternative elimination methods.

Substitution method to solve the equation casesy=1-x x-2y=4 Closing casesThis paper proposes a new method to replace equation (1) with equation (2).

a.$x-2+2x=4$

b.$x-2-2x=4$

Approximately $x-2+x=4$

d.$x-2-x=4$

Answer: A analysis: use substitution to solve the equation casesy=1-x x-2y=4 End cases In this case, we substitute equation (1) into equation (2) to get $x-$$2(1-x)=$$4$, remove the parentheses, and get $x-2+$$2x=4$, so we choose a.

Substituting the elimination method: a system of equations with a coefficient of 1 before an unknown number.

x+y=4 (1).

x-y=2 (2).

To solve this system of equations, one unknown must first be eliminated, and to eliminate an unknown, the sum (difference) of the same unknown in two formulas must be zero, that is to say, the same unknown in both formulas is a positive (or negative), then subtraction, if the same unknown in the two formulas has different signs (i.e., one positive and one negative), then use addition.

Let's eliminate the unknown number y first, and we see that the symbol of the unknown number y in these two equations is different (i.e., one positive and one negative), so we use addition, then we need to use (1) plus (2), how to judge the addition? Let's first write the equation to the left of the equation side by side, and put a plus sign in the equation, x+y+x-y=2x

Then add up the constants to the right of the medium sign of the two formulas, and the result is equal to 6

Since we have previously separated the left and right sides of the two imaginary equations, we now want to restore, that is: 2x=6, then we can dig out the reputation of Zheng to get x=3, and then substitute x=3 into (1) or (2), we now substitute (1) formula, get:

3+y=4 y=1

So the equation is solved.

Addition, subtraction, and elimination method: The method of solving a system of equations with a coefficient of not 1 before an unknown number.

2x+5y=7 (1).

3x+y=4 (2).

We now observe that there are no identical unknowns in the problem, and it is not possible to use the elimination method. So we have to figure out how to make one of the unknowns in the two equations the same, and that's it. If we choose the unknown x, we find that the coefficients before the unknown x in the two equations are 2 and 3 respectively, so we need the least common multiple between 2 and 3.

Since 2 and 3 are both prime numbers, their product 6 is the smallest multiple. Then the whole equation of the first equation must be multiplied by 3, and the whole equation of the second equation must be multiplied by 2, then: (1) multiply the equation by 3 and become:

3*(2x+5y)=7*3, i.e. 6x+15y=21 (3).

2) multiplying by 2 becomes: 2*(3x+y)=4*2 i.e.

6x+2y=8 (4).

Then write equations (3) and (4) together or in two rows.

6x+15y=21

6x+2y=8

After that, we can solve this system of equations, (just follow the method of the example I talked about earlier).

Substitution elimination method is: a mathematical numerical calculation method, which is a simple application of Gaussian elimination method.

The substitution elimination method is to express the unknown of one equation in the system of equations with an algebraic formula containing another unknown, and substitute it into another equation, (it must be another equation, not the equation before the deformation), which eliminates an unknown and obtains a solution. The substitution method is referred to as the substitution method.

1. Example of substitution elimination method:

Change the coefficient of an unknown at the beginning of one of the equations to 1 and substitute it for the other equation. For example: 2x+y=9 2x-y=-1 solution:

Substituting y=9-2x yields:2x-(9-2x=-1x =2, so the solution of the system of equations is x=2y=5.

From the above, it is known that to find the solution of the binary linear equation system, it is through the substitution method to transform the binary linear equation into a univariate equation, and the unknown problem is transformed into a known problem to solve.

2. Ideas: The basic idea of solving equations is to "eliminate the element" - to change the "binary" into "unary". That is to say, the basic idea of solving a system of binary equations is to eliminate the element, and to achieve the elimination element by substitution.

Steps for solving a system of binary linear equations by substitution:

1. Select a binary equation with simple coefficients to deform, and use an algebraic formula containing one unknown to represent another unknown.

2. Substitute the deformed equation into another equation to eliminate the key, eliminate an unknown number, and obtain a unary equation (when substituting, it should be noted that the original equation cannot be substituted, but can only be substituted into another equation without deformation, so as to achieve the purpose of elimination).

3. Solve this unary equation and find the value of the unknown.

4. Substitute the value of the obtained unknown into the deformed equation in 1 to find the value of another unknown.

5. The value of two unknowns is the solution of the system of equations by "{".

6. Finally, check whether the results obtained are correct (substitute into the original equation system for testing, whether the equation satisfies the left = right) and get the difficulty.

Teaching Objective 1: To enable students to understand that the basic idea of solving equations is the idea of elimination. 2. A basic method for students to understand the elimination method is the substitution method, and master the direct substitution elimination method. 3. Through the substitution of the elimination element, students can initially understand the thinking method of transforming the "unknown" into "known" and complex problems into simple problems.

Teaching analysis focus: the elimination process from binary to unary using substitution method. Difficulty:

After using the substitution method to find an unknown value, it is easier to get a simple explanation of which equation to substitute into another unknown value. Breakthrough: Practice more.

Teaching process. 1. Review 1: What is a binary equation? What is a system of binary equations? What is the solution of a system of binary equations?

2. Review the question of the previous lesson: the price of bananas is 5 yuan kg, the price of apples is 3 yuan kg, Xiaohua bought a total of 9 kg and paid 33 yuan. How many kilograms of bananas and apples were bought?

Try to set two unknowns, let's buy bananas x kilograms and buy apples y kilograms, you can list the following two equations: x+y=9 5x+3y=33 and then we get a system of binary linear equations: How is this system of equations solved?

5x+3(9-x)=33By observing the characteristics of the above two equations, it is not difficult to see that the equation is similar to this equation, and the factor 3 is followed by one y and the other is 9-x. So guess y is 9-x, y=9-x? Why?

Then guide the students to observe that it will be regarded as an equation about y, from which it is obtained, y=9-x, and then substitute into , that is, replace y in with 9-x, and get a new equation 5x+3(9-x)=33, solve this equation to get x=3, substitute it into , get y=6, and then find the solution of the system of equations as . From the above, we know that to find the solution of the binary system of equations, we can transform the system of binary equations into a unitary equation by substituting the elimination method, and transform the unknown problem into a known problem. That is to say, the basic idea of solving binary systems of equations is to eliminate the element, and to achieve the elimination element through substitution, and the following is to learn the direct substitution method.

2. Example 1 (see p10) solves the system of equations: Analysis: The equation shows that y and 1-x are equivalent, and the y of the equation can be replaced by 1-x to eliminate y and obtain a linear equation with respect to x.

Emphasis: Imitate the format of the example questions and write the process, and test the oral arithmetic. Variants:

3. Exercise P13: 2(1). 4. Summary 1: What is the basic idea of solving binary linear equations?

2. What are the general steps for solving a system of binary equations? How to test if the logarithmic number of the sum and land is the solution of a certain system of equations.

The concept of addition, subtraction, and elimination is to use the property of the equation to make the absolute value of the coefficient before one of the two unknowns in the equation system equal, and then add or subtract the two equations to eliminate the unknown, so that the equation contains only one unknown and can be solved.

Use the properties of the equation to make the absolute value of the coefficient before one of the two equations in the equation equal, and then add or subtract the two equations to eliminate the unknown, so that the equation contains only one unknown and can be solved. This method of adding or subtracting an unknown number from both sides of two equations is called addition, subtraction, and subtraction, which is also called Gaussian elimination method because it was proposed by the mathematician Gauss.

The steps of solving a system of binary equations by addition and subtraction: using the basic properties of the equation, the coefficients of an unknown number in the original system of equations are reduced to the form of equal or opposite numbers; Then use the basic properties of the equation to add or subtract the two deformed equations, eliminate an unknown number, and obtain a unary equation (be sure to multiply both sides of the equation by the same number, do not multiply only one side, and then use subtraction if the unknown coefficients are equal, and add if the unknown coefficients are opposite to each other).

Solve this unary equation and find the value of the unknown; Substituting the value of the obtained unknown into any one of the equations in the circle of the original equation to find the value of another unknown; The value of two unknowns is the solution of the system of equations by "{"; Finally, check whether the obtained result is correct (substitute the original equation system to test, whether the equation satisfies the number on the left = the number on the right).

Introduction to Mathematics:

Mathematics is the study of concepts such as quantity, structure, change, space, and information. Mathematics is a general means for humans to strictly describe the abstract structure and pattern of things, and can be applied to any problem in the real world, and all mathematical objects are inherently artificially defined. In this sense, mathematics belongs to the formal sciences, not the natural sciences.

Mathematicians and philosophers of different voices have a range of opinions about the exact scope and definition of mathematics. Mathematics plays an irreplaceable role in the development of human history and social life, and it is also an indispensable basic tool for learning and researching modern science and technology. In ancient China, mathematics was called arithmetic, also known as arithmetic, and finally changed to mathematics.

Arithmetic in ancient China is one of the six arts (called "number" in the six arts).

Mathematics originated in the early production activities of human beings, and the ancient Babylonians had accumulated a certain amount of mathematical knowledge since ancient times and could apply practical problems. From the perspective of mathematics itself, their mathematical knowledge is only obtained through observation and experience, and there is no comprehensive conclusion and proof, but it is also necessary to fully affirm their contributions to mathematics.

Equation. , an unknown number is called a yuan, an equation containing one unknown is called a univariate equation, and an equation containing two unknowns is called a binary equation.

An equation with three unknowns is called a ternary equation, and so on.

N n-element equations form an n-element system of equations (n-element simultaneous equations), and to solve the system of equations, try to gradually reduce the unknowns in an equation, this process is called elimination.

There are substitution and subtraction of the elimination method.

The substitution method is: assuming that the unknown number A can be represented by the formula containing the unknown number B, then the formula containing the unknown number B can be used to replace the unknown number A, which is equivalent to eliminating the unknown number A.

If you're looking for an example, please ask.