The concept of (1): the unknown number of one equation in the equation group is expressed by an algebraic expression containing another unknown number, then it is substituted into another equation, and an unknown number is eliminated to obtain a linear equation with one variable, and finally the solution of the equation group is obtained. This method of solving equations is called substitution elimination method, or substitution method for short.
(2) The step of solving binary linear equations by substitution method.
① A binary linear equation with simple coefficients is selected for deformation, and an algebraic expression containing an unknown number is used to represent another unknown number;
(2) Substitute the deformed equation into another equation, eliminate an unknown number, and get a linear equation (when substituting, be careful not to substitute into the original equation, only substitute into another equation without deformation, so as to achieve the purpose of elimination);
③ Solve this one-dimensional linear equation and get the unknown value;
(4) Substituting the obtained unknown values into the deformation equation in (1),
Find the value of another unknown;
⑤ Simultaneous two unknowns with "{"are the solutions of equations;
⑥ Final test (substitute the original equation to test whether the equation satisfies left = right).
Example:
{x-y=3 ①
{3x-8y=4②
X=y+3③ Starting from ①.
③ Substitute into ② to get it.
3(y+3)-8y=4
y= 1
Bring y= 1 into ③.
Get x=4.
Then: the solution of this binary linear system of equations
2) Addition, subtraction and elimination methods
The concept of (1): When the coefficients of the unknowns of two equations in an equation are equal or opposite, the unknowns are eliminated by adding or subtracting the two sides of the two equations, so that the binary linear equation is transformed into a univariate linear equation, and finally the solution of the equation is obtained. This method of solving equations is called addition, subtraction and elimination, or addition and subtraction for short.
(2) Add and subtract steps to solve binary linear equations.
① Using the basic properties of the equation, the coefficient of an unknown quantity in the original equation is transformed into an equal or opposite number;
(2) Using the basic properties of the equation, add or subtract two deformation equations to eliminate an unknown number, and get a linear equation (both sides of the equation must be multiplied by the same number, and it is forbidden to multiply only one side. If the unknown coefficient is equal, it will be subtracted, and if the unknown coefficient is opposite, it will be added);
③ Solve this one-dimensional linear equation and get the unknown value;
(4) substituting the obtained unknown value into any equation in the original equation set,
Find the value of another unknown;
⑤ Simultaneous two unknowns plus "{"is the solution of the equations.
⑥ Finally, check whether the result is correct (substitute into the original equations and check whether the equations satisfy left = right).
For example:
The first equation ① and the second equation ② are called.
①×2 to obtain ③
10x+6y= 18
③-② Obtain:
10x+6y-( 10x+5y)= 18- 12
3) Alternative methods
When solving a mathematical problem, we regard a formula as a whole and replace it with a variable, thus simplifying the problem. This is called substitution method. The essence of substitution is transformation, the key is to construct elements and set elements, and the theoretical basis is equivalent substitution. The purpose is to change the research object, move the problem to the knowledge background of the new object, standardize non-standard problems, simplify complex problems and become easy to deal with.
Substitution method is also called auxiliary element method and variable substitution method. By introducing new variables, scattered conditions can be linked, implicit conditions can be revealed, or conditions can be linked with conclusions. Or turn it into a familiar form to simplify complicated calculation and derivation.
It can transform high order into low order, fraction into algebraic expression, irrational expression into rational expression, transcendental expression into algebraic expression, and has a wide range of applications in the study of equations, inequalities, functions, sequences, triangles and other issues.
take for example
(x+y)/2-(x-y)/3=6①
3(x+y)=4(x-y)②
Solution: let x+y be a and X-Y be b.
So, the original equation becomes
a/2-b/3=6③
3a-4b=0 ④
Solution:
a=24
b= 18
Therefore:
x+y=24
x-y= 18
The solution of the equation is:
x= 2 1
y= 3 .