How to solve a binary equation using addition, subtraction, and elimination

Using the basic properties of the equation, the coefficient of an unknown number in the original equation system is 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 original equations 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 results obtained are correct (substituted into the original equation system for testing, whether the equation satisfies the left = right).

Examples: 1 3x+2y=7 ①

5x-2y=1 ②

Solution: 3x+5x)+2y+(-2y)=(7+1).

8x=8 ∴ x=1

Substitute x : 3x+2y=7 3 1+2y=7 2y=4 y=2

x=1 y=2

It's hard to say, but you should have it in math, and look at the list more.

The steps of the binary linear equation addition and subtraction method are as follows:

The general method of solving binary (ternary) equations is to substitute the elimination method and the addition and subtraction elimination method.

1 Solution: 1) The substitution elimination method is to express the unknown number of one of the equations in the system of equations with an algebraic formula containing another unknown, and substitute it into another equation to eliminate another unknown to obtain a solution. The substitution method is referred to as the substitution method.

2) The addition, subtraction, and elimination method uses 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. This method of solving a system of binary equations is called addition, subtraction, and subtraction.

The general steps to eliminate elements by addition and subtraction are:

In a system of binary equations, if there are coefficients of the same unknown number that are the same (or opposite to each other), it can be directly subtracted (or added) to eliminate an unknown;

In a system of binary linear equations, if there is no such thing as in, an appropriate number can be selected to multiply both sides of the equation so that the coefficients of one of the unknowns are the same (or opposite to each other), and then the two sides of the equation are subtracted (or added) respectively to eliminate an unknown number to obtain a unary linear equation;

Solve this unary equation;

The solution of the unary linear equation is substituted into the equation with the relatively simple coefficients of the original equation, and the value of another unknown is obtained.

The values of the two unknowns are concatenated in curly brackets, which is the solution of a system of binary linear equations.

2. Thought: "Elimination of elements", that is, the transformation of "duality" into "unity", this method embodies the idea of naturalization in mathematical research, specifically the transformation of "new knowledge" into old knowledge, the transformation of "unknown" into "known", and the transformation of "complex problems" into "simple problems".

Convert a system of binary linear equations into a unitary equation so that one unknown can be solved first and then try to find another unknown. This idea of reducing the number of unknowns from more to less and solving them one by one is called the idea of eliminating the element.

The specific transformation method is to use the "substitution elimination method" or "addition and subtraction elimination method" to eliminate one unknown number from the two unknowns in the binary linear equation system to obtain the unary equation, so as to achieve the elimination and then solve the problem. Here's an example:

1. Use the substitution method to quickly evaluate:

In an equation of a system of binary linear equations, an unknown number is represented by a formula containing another unknown, and then substituted into another equation to achieve elimination, and then the solution of this system of binary linear equations is obtained.

2. Use addition and subtraction to evaluate quickly.

When the coefficients of the same unknown in two binary equations are opposite or equal, add or subtract the two sides of the two equations respectively to eliminate this unknown and obtain a unitary equation, which is called addition, subtraction, and subtraction.

Rational use of this idea, in the evaluation problem can also get twice the result with half the effort.

Example 3If 4x+5y=10, and 5x+4y=8, then.

From the meaning of the title: from 9x+9y=18 i.e.: x + y= 2

Result: x y=-2

So -1 comment: If you directly form a system of equations with 4x+5y=10 and 5x+4y=8, find the solution of the system of equations, and then substitute the solution into the evaluation. This is not only computationally intensive, but also error-prone.

If you carefully analyze the evaluated formula, you can consider using addition and subtraction to quickly obtain the values of x+y and x-y, so this problem is familiar with the mathematical ideas in the binary linear equations, mainly the "elimination" idea of exponentials, that is: there are two unknowns in the binary equations, if one of the unknowns is eliminated, the binary linear equations are converted into unary equations, so that one unknown can be solved first, and then another unknown can be solved. This idea of reducing the number of unknowns from more to less and solving them one by one is called the idea of eliminating the element.

The specific transformation method is to use the "substitution elimination method" or "addition and subtraction elimination method" to eliminate one unknown number from the two unknowns in the binary linear equation system to obtain the unary equation, so as to achieve the elimination and then solve the problem. Here's an example:

1. Use the substitution method to quickly evaluate:

In an equation of a system of binary linear equations, an unknown number is represented by a formula containing another unknown, and then substituted into another equation to achieve elimination, and then the solution of this system of binary linear equations is obtained. This method is called the substitution elimination method, referred to as the substitution method.

2. When the coefficients of the same unknown in the addition and subtraction of two binary equations are opposite or equal, the two sides of the two equations are added or subtracted respectively to eliminate this unknown and obtain a unary equation, which is called the addition and subtraction method, referred to as addition and subtraction.

Specifically, it should be called a system of binary equations, because two unknowns require two equations to solve the value of the unknown number.

For example: (1): 2x+3y=8, (2):

3x+2y=9;(1) Multiply 3 to get 6x+9y=24; (2) Multiply 2 to get 6x+4y=18; then use 6x+9y-(6x+4y)=24-18; 5y=6; Solution: y=6 5; Then substitute y=6 5 into 2x+3y=8, and the solution is: x=11 5.

Or (1) multiply 3 by 2.

(1) Concept: When the coefficients of an unknown number of two equations in the equation system are equal or opposite to each other, the two sides of the two equations are added or subtracted to eliminate the unknown, so that the binary equation is transformed into a one-dimensional equation, and finally the solution of the equation system is obtained, and the method of solving the equation system is called addition, subtraction and subtraction, referred to as addition and subtraction.

2) Steps to solve a system of binary equations by addition and subtraction.

Using the basic properties of the equation, the coefficient of an unknown number in the original equation system is 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 original equations 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 into the original equation system for testing, whether the equation satisfies the number on the left = the number on the right).

Examples: 1 3x+2y=7 ①5x-2y=1 ②

Solution: 3x+5x)+2y+(-2y)=(7+1).

8x=8 x=1

Substituting x into :3x+2y=7

3×1+2y=7

2y=4 y=2

x=1y=2

Listen well in class, go to school and ask the teacher.

Addition, subtraction, and elimination are actually the addition and subtraction of equations, making it more convenient to solve multiple systems of equations! If you understand this, you can basically learn the addition and subtraction of the meta-method.

Addition, subtraction, and elimination method: When the coefficients of an unknown number of two equations in the equation are equal or opposite to each other, the two sides of the two equations are added or subtracted to eliminate the unknown, so as to turn the binary equation into a one-dimensional equation, and finally obtain the solution of the system of equations.

Steps. Using the basic properties of the equation, the coefficients of an unknown number in the original equation cave group 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 original equations 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 results obtained are correct (substituted into the original equation system for testing, whether the equation satisfies the left = right).

1) Subtract the second equation from the first equation to obtain: (3x-3x)+2y-(-2y)=9-3

4y=6y=6/4

Approximation. y=3/2

Substitute y=3 2 into 3x-2y=3.

3x-2*3/2=3

3x=6x=2

2) Add the first equation to the second equation to obtain: (6x+3x)+(4y+4y)=37-4

9x=33x=33/9

Put x=33 9

Substitute 3x+4y=-4.

4y=-15

y=-15/4

3) Put 6m-3n = 15

Divide by 3 on both sides

Get: 2m-n=5

Then multiply 2m-n=5 by 8 on both sides to get :

16m-8n=40

Put 9m + 8n = 10

with 16m-8n = 40

Summing: (9m+16m)+(8n-8n)=40+1025m=50

m=2 is substituted into 2m-n=5.

4-n=5n=-1

4) Add the two equations to obtain: (x-3) 2+(x-3) 2+ (y-2) 3-(y-2) 3 =6

x-3+x-3)/2=6

Multiply the dust on both sides by 2

2x-6=12

2x=18x=9

Substitute x=9.

x-3)/2+(y-2)/3=4

3+(y-2)/3=4

y-2)/3=1

y-2=3y=5

I've been fighting for a long time! (Hopefully, it will work for you!) ~

2x+5y=15

3x+8y=-1

Multiply the two sides of the first equation by 3: 6x+15y=45

The second equation is multiplied by 2 on both sides: 6x+16y=-2 (that is, let the coefficient of x be judged by the cherry blossoms, etc., so look for the least common multiple to change the number of rounds, so as to eliminate x).

Subtract the two equations y=-47

x=125

p> uses the properties of the equation to make the absolute value of the coefficient before one of the two unknowns in the system of equations equal; This method of solving a system of binary equations is called addition, subtraction, and elimination, in which the equation contains only one unknown and is solved, and then the two equations are added or subtracted to eliminate the unknown.

Question 1 x 3-y 4 = 3 ......a

x/2+y/3=13……b

a*4+b*3 gets 17x 6=51, so x=18, y=12 The second question is a bit messy, is it the original equation ax+by=26, &cx+y=6, and the nickname is correct x=4, &y=-2

Xiao Yin wrote C incorrectly......

x=4, &y=-2 substituting cx+y=6 to get 4c-2=6 to get c=2, according to ax+by=26, to get 4a-2b=26......m7a+3b=26……n

m*3+n*2 gives 26a=26*5, so a=