How to solve binary solution equations ? How to solve binary equations

Answer: The elimination method is to convert the duality into a unitary method, which is generally the addition and subtraction elimination method or the substitution elimination method. j

1. Substitution of the elimination method.

1) Concept: An unknown of an equation in a system of equations is expressed by an algebraic formula containing another unknown, substituted into another equation, an unknown is eliminated, a unary equation is obtained, and finally the solution of the system of equations is obtained. This method of solving a system of equations is called the substitution elimination method, or substitution method for short.

2) Substitution method to solve the steps of a system of binary linear equations.

A binary linear equation with simple coefficients is selected to deform and another unknown is represented by an algebraic formula containing one unknown.

Substitute the deformed equation into another equation, eliminate an unknown number, and obtain a unary one-dimensional 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.) ）

Solve this unary equation and find the value of the unknown;

Substituting the value of the obtained unknown into the deformed 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 results obtained are correct (substituted into the original equation system for testing, whether the equation satisfies the left = right).

Example: x-y=3 3x-8y=4

Substituting x=y+3 gives 3(y+3)-8y=4 y=1 so x=4 then: the solution of this system of binary linear equations {x=4 {y=1

2. Addition, subtraction, and elimination method.

1) Concept: 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, the method of solving the system of equations 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 results obtained are correct (substituted into the original equation system for testing, whether the equation satisfies the left = right).

Example: x-y=3 3x-8y=4

1) Multiply by 3 to get 3x-3y=9 (4).

Use Eq. (4) - 2) to get 5y=5 so y=1

Substituting (1) gives x=4

1. Substitution of the elimination method.

For example, the system of equations for solving the base spike is changed to x+y=5

6x+13y=89②

Solution: From x=5-y

Put substitution in, got.

6(5-y)+13y=89

i.e. y=59 7

Substituting y=59 7 into x=5-59 7 gives x=-24 7

x=-24/7

y=59 7 is the solution of the system of equations.

We call this method of eliminating an unknown number by "substitution" to find the solution of the system of equations (elimination by substitution).

2. Addition, subtraction, and elimination method.

Example: Solve a system of equations: x+y=9

x-y=5②

Solution: +Get 2x=14

That is, x=7 substituting x=7 into , we get 7+y=9

solution, get: y=2

x=7y=2 is the solution of the system of equations.

This method of solving a system of binary equations is called elimination by addition-subtraction, or addition and subtraction for short.

How to solve binary equations, detailed solutions.

Binary Linear Equations?

Use the Root Finding Formula Method! ax code destroy+bx+c=0(a≠0),x=(-b laugh pei(b 2-4ac)) 2a.

For example, the solution of a quadratic equation 2x +3x+1=0 is x=(-3 (3 2-4 2 1)) 2 2 The solution is only x1=-1, x2=-1 2

1.Transformation: Transform this unary quadratic equation into the form ax 2+bx+c=0 (i.e., the general form of the unary quadratic equation) into the general form 2

Shift: The constant term is moved to the right of equation 3Coefficient: 1:

ax 2+bx+c=a(x+b 2a) 2+(4ac-b 2) 4a=a[(x+m) 2-n 2]=a(x+m+n)*(x+m-n) Example: Solve the equation 2x 2+4=6x 1 2x^2-6x+4=0 2.

x^2-3x+2=0 3. x^2-3x=-2 4. x^2-3x+ （

Add 3 and a half squared, and -2 also add 3 and a half squared to make both sides of the equation equal) 5(a 2+2b+1=0 i.e. (a+1) 2=0) 6 7.

x1=2 x2=1 (a quadratic equation usually has two solutions, x1 x2).

This paragraph of the quadratic function with method skills.

y=ax&sup, which is often used in solving equations, inequalities, and functions, is detailed below: First of all, it is clear that the matching method is to write about two numbers (or algebraic slippery reeds, but these two must be flat) in the form of (a+b) square or (Lie type a-b) squared: (a+b) squared to obtain (a+b) 2=a 2+2ab+b 2 The letter is in the form of (a+b) squared, and it is necessary to have a 2,2ab,b 2 After selecting the objects you want to match (that is, A 2 and B 2, this is the core, you must have these two objects, otherwise you can't use the recipe formula), add and add, for example:

The original formula is a 2+ b 2 solution: a 2+ b 2 = a 2+ b 2 +2ab-2ab = a 2+ b 2 +2ab)-2ab = a + b) 2-2ab Another example: The original formula is a 2+ 2b 2 solution:

A 2+2b 2 = a 2+ b 2 + b 2 +2ab-2ab = a 2+ b 2 +2ab)-2ab+ b 2 = a+b) 2-2ab+ b 2 This is how it is matched, Note: If there is a coefficient before a or b, it is considered as a or part of b, e.g. 4a 2 is considered (2a) 2 9b 2 is considered (a 29b 2).

by 4x(x2+y2)-4x=0

Get: x(x 2+y 2-1)=0

So: x=0, or x2+y2=1 -- the first set of equations) is given by 4y(x2+y2)+4y=0

Get: y(x 2+y 2+1)=0

So: y=0 --the second set of equations), note: x 2+y 2+1=0 has no real root, so clear it from the second set of equations.

The next step is to combine group 2 and group 1 to get: y=0, x=0,-- which is the solution of group 1.

Or y=0 and x 2+y 2=1 are combined, and we get:

y=0，x^2=1

So: y=0, x=1,-- this is the second set of solutions.

Or y=0, x=-1,-- which is the 3rd set of solutions.

is composed of 4y(x2+y2)+4y=0

We get y(x 2+y 2+1)=0, x 2+y 2+1>0, so y=0

Substituting 4x(x 2+y 2)-4x=0, you get.

4x 3-4x=0, the solution is x=0, or soil 1

May I?

Finding Extrema for Binary Functions:

First, find the partial derivatives of the first order: z x, z y so that z x=0, z y=0 solve the system of equations to find the stationary points (x ,y), x ,y )

Then find the second-order partial derivatives: z x , z x y, z y so that z x =a, z x y=b, z y =c at the station, and substitute the station into z x, z x y, z y, and find a, b, c

Discriminant: p=b -ac

When p<0 and a<0, the function at the station point obtains a maximum.

When p<0 and a>0, the function at the station point obtains a minimum.

When p>0 is not an extreme point, when p=0 is not an extreme point, it is necessary to judge separately.

1. Substitution of the elimination method.

Example: Solve a system of equations: x+y=5

Tell your friends to change the sock sentence 6x+13y=89

Solution: From x=5-y

Put substitution in, got.

6(5-y)+13y=89

i.e. y=59 7

Substituting y=59 7 into x=5-59 7

i.e. x=-24 7

x=-24/7

y=59 7 is the solution of the system of equations.

We call this method of eliminating an unknown number by "substitution" to find the solution of the system of equations (elimination by substitution).

2. Addition, subtraction, and elimination method.

Example: Solve a system of equations: x+y=9

x-y=5②

Solution: Feast + get 2x=14

That is, x=7 substituting x=7 into , we get 7+y=9

solution, get: y=2

x=7y=2 is the solution of the system of equations.

This method of solving a system of binary equations is called elimination by addition-subtraction, or addition and subtraction for short.

Solve like a one-dimensional equation.

Turn it into a one-dimensional equation.

Putting the second equation 2, we get n-m=5 2 and we get n=5 2+m

Substituting n into the first formula, finding m, and then finding n.

16a+4b=－62①

36a+6b=－75②

3. 48a+12b=－186③

2. 72a+12b=－150④

Get. 24a=36,a=

Substitution. 16×

24+4b=－62

4b=－86

b= so a=, b=

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