There are two unknowns (generally set as x and y), and the number of items containing unknowns is 1. Equations like this are called binary linear equations. For example, X+Y = 24 are binary linear equations.

Key point explanation:

In the (1) equation, "yuan" means unknown number, and "binary" means that there are only two unknowns in the equation.

(2) "The number of unknowns is 1" means that the number of items (monomials) containing unknowns is 1. For example, the number of xy is 2, so the equation

6 xy+9 = 0 is not a binary linear equation.

(3) The left and right sides of a binary linear equation must be algebraic expressions. If the left side of the equation is not an algebraic expression, it is not binary.

Quadratic equation.

(4) to judge whether an equation is a binary linear equation, it is generally converted into the form of AX+BY+C = 0, and then judged according to the definition, for example.

For example, 2x+4y = 3+2x is not a binary linear equation, because the original equation is changed to 4y = 3 by shifting the term, which does not conform to the binary linear equation.

Form.

Knowledge point 2: the solution of binary linear equation

The values of two unknowns that can make the values of the left and right sides of the binary linear equation equal are called the solutions of the binary linear equation. Because there is more than one unknown to make the left and right sides of a binary linear equation equal, every binary linear equation has countless groups of solutions.

For example,,,,, are all solutions of the binary linear equation x+y = 3. We call such an equation with countless groups of solutions an indefinite equation.

Key point explanation:

(1) The values of two unknowns that make the left and right sides of a binary linear equation equal (every solution of a binary linear equation is a pair of values, but it is not.

Is a numerical value), that is, the solution of binary linear equation must be combined with "{",for example, it is the solution of binary linear equation x+y = 2.

(2) Among countless solutions of binary linear equation, the values of two unknowns are interrelated and corresponding to each other. Which is the value of one of the unknowns.

After confirmation, the value of another unknown is also confirmed and unique.

Knowledge point 3: the concept of binary linear equations

Two binary linear equations with the same unknowns are combined into one binary linear equation group.

For example, they are all binary linear equations.

In addition, the equations that make up the equations need not contain two unknowns at the same time.

For example, it is also a system of binary linear equations.

Knowledge point 4: Solution of binary linear equations

Generally speaking, the common solution of two equations of binary linear equations is called the solution of binary linear equations.

Key point explanation:

The solution of the (1) equation should be simultaneous with the brackets, for example, instead of x = 9 and y = 4.

(2) Generally speaking, there is only one solution for binary linear equations, but there are also special cases, such as equations without solution, equations.

There are countless solutions.

(3) When testing whether a set of numbers is the solution of binary linear equations, we must substitute this set of numbers into each equation group to see if it is.

Satisfy every equation. Only when this set of numbers satisfies all equations in the system of equations, this set of numbers is the solution of the original system of equations, otherwise it is not.

Knowledge point 5: Exclusion method

1. elimination idea: there are two unknowns in binary linear equations. If one of the unknowns is eliminated, then the binary linear equation is

By transforming it into a once-in-a-lifetime equation that we are familiar with, we can find one unknown first and then another unknown. This kind.

The idea of solving the unknowns one by one is called elimination thought.

2. The basic idea of elimination method: unknowns are variable and few.

3. The basic method of elimination: the binary linear equations are transformed into the univariate linear equations.

Knowledge point 6: substitution exclusion method

1. Substitution elimination method is one of the two basic methods for solving equations. The substitution elimination method is one of the equations that take an unknown number.

It is expressed by the algebraic expression of another unknown, and then it is substituted into another equation to eliminate an unknown and transform it into a binary linear equation system.

Solve one-dimensional linear equation. This method of solving binary linear equations is called substitution elimination method, which is called substitution method for short.

2. method of substitution's general steps to solve binary linear equations:

(1) Choose an equation with relatively simple coefficients from the equations, and use one unknown in this equation as the algebraic expression of another unknown.

Display;

(2) Substituting the deformed relation into another equation, eliminating the unknowns and obtaining a linear equation;

(3) Solve this one-dimensional linear equation and find an unknown value;

(4) Substituting the obtained value of the unknown quantity into the deformation relation to obtain the value of another unknown quantity;

(5) The values of two unknowns are combined with the symbol "{"to write the solution of the equation.

Key point explanation:

(1) When solving binary linear equations by substitution method, we should first observe the characteristics of each coefficient and try to choose the simple one after deformation or substitution.

Simple equation deformation;

(2) The deformed equation can no longer be substituted into the original equation, but can only be substituted into another equation in the original equation set;

(3) Be good at analyzing the characteristics of equations and looking for simple solutions. For example, if an unknown number and its coefficient are regarded as a whole with another number.

Substituting the algebraic expression of the unknown into another equation, or directly substituting one equation into another equation, is called the whole.

Substitution method. Integral substitution method is one of the commonly used methods to solve binary linear equations, and its application can make the operation simple and improve the operation speed.

And accuracy.

Knowledge point 7: addition, subtraction and elimination method

1. Addition, subtraction and elimination method is one of the basic methods to solve binary linear equations. The method of addition, subtraction and elimination is to eliminate two equations by addition (or subtraction).

An unknown, binary linear equations are transformed into univariate linear equations to solve. This solution is called addition, subtraction and elimination, or addition and subtraction for short.

2. The general steps of solving binary linear equations by addition and subtraction:

Two equations in the (1) equations can be multiplied by an appropriate number if the coefficients of the same unknown are neither opposite nor equal.

On both sides of an equation or two equations, the coefficient of an unknown in the two equations is opposite or equal;

(2) Add and subtract the two sides of the two equations respectively (subtraction at the same time, addition at the opposite time) to eliminate an unknown and get a quadratic formula.

Cheng;

(3) Solve this one-dimensional linear equation and get the value of one of the unknowns;

(4) Substituting the obtained value of the unknown quantity into the equation with relatively simple coefficient in the original equation to obtain the value of another unknown quantity;

(5) The obtained values of two unknowns are combined with the symbol "{"to write the solution of the equation.

Key point explanation:

Generally speaking, the selection methods of addition, subtraction and elimination are:

(1) Eliminate the unknown with smaller absolute value of selection coefficient;

(2) The absolute value of an unknown is equal. If the symbols are different, they are eliminated by addition. If the signs are the same, they are eliminated by subtraction.

(3) When the unknown coefficients have multiple relations, directly deform one of the equations to make the absolute values of the coefficients equal, and then eliminate them by addition and subtraction.

Yuan;

(4) When the coefficients of the same unknown are not equal, find the least common multiple of an unknown and deform the two equations at the same time.

Convert it into coefficients with the same absolute value and then solve it by addition and subtraction.

When solving equations by addition and subtraction, we should pay attention to the following points: ① When dealing with the deformation of an equation, everything should be expanded by the same multiple; ② The left and right terms of the two equations should be added or subtracted at the same time.

Third, the guidance of legal methods.

1. Solution of integer solution of binary linear equation: In general, a binary linear equation has countless integer solutions. When solving this kind of problems, an unknown number is represented by algebra, and then the corresponding solutions are obtained one by one according to the conditions.

2. Method of judging binary linear equations: two binary linear equations with the same unknowns are combined into a binary linear equation group, and judging whether an equation is a binary linear equation group depends on whether it meets the following two conditions: (1) whether there are two unknowns in the whole equation group; (2) See if the number of items with unknowns is 1.

3. To test whether a pair of numbers is the solution of a binary linear equation system, the common methods are as follows: substituting the pair of numbers into each equation of the equation system respectively, and only when the pair of numbers satisfies all equations can it be said that the pair of numbers is the solution of this equation system; Otherwise, if the pair of values does not satisfy any equation, then it is not the solution of the equation.

4. Problems needing attention in solving binary linear equations by substitution method and addition and subtraction method:

(1) When the equations contain an algebraic expression that one unknown represents another, the substitution method is relatively simple;

(2) If the coefficient of the unknown quantity in the equation is 1 (or-1), it is simpler to choose the equation with the coefficient of 1 (or-1) for deformation.

(3) When the coefficients of two equations are the same or opposite, it is easy to add, subtract and eliminate;

(4) If the coefficient of the same unknown quantity in the two equations is multiple, it can be converted into the type of (3) by using the properties of the equations, and you can choose addition and subtraction.

The elimination method is relatively simple;

(5) If the absolute values of the coefficients of the same unknown quantity in two equations are not equal, a group of coefficients (the least common multiple is smaller) should be selected.

A set of coefficients), find their least common multiple, and then transform the original equation to make the absolute values of these coefficients of the new equation equal.

(all equal to the least common multiple of the original coefficient), and then add, subtract and eliminate;

(6) For complex binary linear equations, we should simplify them first (remove denominator, brackets, merge similar terms, etc.). ). Usually, every equation should be simplified.

Arrange it in the form of unknown term on the left and constant term on the right of the equation, and then consider adding, subtracting and eliminating.