Binary linear equations

1. Binary linear equation: contains two unknowns, and the degree of the unknowns is 1. Such an equation is a binary linear equation. Note: Generally speaking, a binary linear equation has countless solutions.

2. Binary linear equations: two binary linear equations are put together.

3. Solution of binary linear equations: The values of two unknowns that make the left and right sides of two equations of binary linear equations equal are called the solutions of binary linear equations. Note: Generally speaking, there is only one solution (the common * * * solution) for binary linear equations.

4. Solutions of binary linear equations;

(1) substitution elimination method; (2) addition, subtraction and elimination;

(3) Note: It is the key to judge how to solve the problem simply.

5. Application of linear equation. ※:

(1) For an application problem, the more unknowns are set, the easier it may be to establish the equation, but the more troublesome it may be to solve the equation, otherwise it will be "difficult to set and easy to solve";

(2) For equations, if the number of equations is equal to the number of unknowns, the value of the unknowns can generally be obtained;

(3) For equations, if the number of equations is one less than the number of unknowns, then the value of the unknowns cannot be found, but the relationship between any two unknowns can always be found.

One-dimensional linear inequality (group)

1. inequality: a formula connecting two algebraic expressions with inequality symbol >

2. The basic properties of inequality:

The basic property of inequality is 1: add (or subtract) the same number or the same algebraic expression on both sides of inequality, and the direction of inequality remains unchanged;

The basic properties of inequality 2: both sides of inequality are multiplied (or divided) by the same positive number, and the direction of inequality remains unchanged;

The basic property of inequality 3: both sides of inequality are multiplied by (or divided by) the same negative number, and the direction of inequality should be changed.

3. Solution set of inequality: the value of the unknown quantity that can make the inequality hold is called the solution of this inequality; The set of all solutions of an inequality is called the solution set of this inequality.

4. One-dimensional linear inequality: an inequality with only one unknown number, degree 1 and coefficient not equal to zero is called one-dimensional linear inequality; Its standard form is ax+b > 0 or ax+b < 0, (a≠0).

5. Solution of one-dimensional linear inequality: The solution of one-dimensional linear inequality is similar to the solution of one-dimensional linear equation, but we must pay attention to the application of inequality property 3; Note: when representing the solution set of inequality on the number axis, we should pay attention to the empty circle and real point.

6. One-dimensional linear inequality group: an inequality group consisting of several one-dimensional linear inequalities with the same unknown number is called one-dimensional linear inequality group; Note: ab > 0? ? Or;

ab B。

9. Several important judgments:,,

Multiplication and division of algebraic expressions

1. Multiplication with the same base: am? An=am+n, constant radix, exponential addition.

2. The power of the power and the power of the product: (am)n=amn, the base is unchanged, multiplied by the index; (ab)n=anbn, and the power of the product is equal to the product of the power of each factor.

3. Multiplication of monomial: multiplication of coefficients, multiplication of the same letters, and the letters contained in only one factor are written together with the index in the product.

4. Multiply the term with the polynomial: m(a+b+c)=ma+mb+mc, multiply each term of the polynomial by the term, and then add the products.

5. Multiplication of polynomials: (a+b)? (c+d)=ac+ad+bc+bd, multiply each term of another polynomial by each term of this polynomial, and then add the products.

6. Multiplication formula:

(1) square difference formula: (a+b)(a-b)= a2-b2, and the product of the sum of two numbers and the difference between the two numbers is equal to the square difference between the two numbers;

(2) Complete square formula:

(1) (A+B) 2 = A2+2AB+B2, and the square of the sum of two numbers is equal to the sum of their squares plus twice their product;

(2) (a-b) 2 = A2-2AB+B2, and the square of the difference between two numbers is equal to the sum of their squares, minus twice their product;

③ (a+b-c) 2 = A2+B2+C2+2ab-2ac-2bc, omitted. ※ 。

7. Formula:

(1) If the quadratic trinomial x2+px+q is completely flat, there is a relation:

(2) The quadratic trinomial ax2+bx+c can always be changed into the form of a(x-h)2+k after the formula, and a(x-h)2+k can be used. ※

① The symbol of ax2+bx+c value can be judged; ② When x=h, the maximum (or minimum) value k of ax2+bx+c can be found.

(3) Note: ※:

8. same base powers's division: am÷an=am-n, constant base, exponential subtraction.

9. Zero exponent and negative exponent formulas:

( 1)A0 = 1(a≠0)； A-n=，(a≠0)。 Note: 00, 0-2 is meaningless;

(2) For negative exponent, numbers less than 1 can be recorded by scientific notation, such as 0.0000201= 2.01×10-5.

10. monomial division: coefficient division, division with letters, only the letters contained in the division formula, together with its exponent, are taken as a factor of quotient.

1 1. Polynomial divided by monomial: first divide the terms of polynomial by monomial, and then add the obtained quotients.

12. Polynomial divided by polynomial: factorization first and then reduction or vertical division. Note: Divided remainder = divided? ※? Business style.

13. Algebraic expression mixed operation: multiply first, then multiply and divide, and finally add and subtract. If there are parentheses, use the number of parentheses first.

Line segments, angles, intersecting lines and parallel lines

A-level concept of geometry: (requires deep understanding and skillful use, mainly used for geometric proof)

1. Definition of angle bisector:

A ray divides an angle into two equal parts. This ray is called the bisector of an angle (pictured).

Example of geometric expression:

(1) ∵OC ∠AOB.

∴∠AOC=∠BOC

(2)AOC =∠BOC

∴OC is the bisector of ∞∠AOB.

2. The definition of the midpoint of the line segment:

Point C divides the line segment AB into two equal line segments, and point C is called the midpoint of the line segment.

Example of geometric expression:

(1) ∵C is the midpoint of AB.

∴ AC = BC

(2)∫AC = BC

∴C is the midpoint of AB

3. Equality axiom: (as shown in the figure)

(1) Equal amount plus equal amount and equal; (2) Equal amount minus equal amount difference;

(3) Equal amount and equal amount; (4) Equal amounts of the same components are equal.

( 1)∫AC = DB

∴AC+CD=DB+CD

That is, AD = BC.

(2)AOC =∠DOB

∴∠AOC-∠BOC=∠DOB-∠BOC

That is, ∠AOB=∠DOC.

(3)∫∠BOC =∠GFM

∫∠AOB = 2∠BOC。

∠EFG=2∠GFM

∴∠AOB=∠EFG

(4)∫AC = AB，EG= EF

AB = EF

∴AC=EG

4. Equivalent substitution: examples of geometric expressions:

∫a = c

b=c

Example of geometric expression ∴ a = b:

∫a = c b = d

∫c = d

Example of geometric expression ∴ a = b:

∫a = c+d

b=c+d

∴a=b

5. The important properties of complementary angle:

The complementary angles of the same angle or the same angle are equal.

Example of geometric expression:

∵∠ 1+∠3= 180

∠2+∠4= 180

∫∠3 =∠4

∴∠ 1=∠2

6. The important properties of complementary angle:

The complementary angles of the same angle or the same angle are equal.

Example of geometric expression:

∵∠ 1+∠3=90

∠2+∠4=90

∫∠3 =∠4

∴∠ 1=∠2

7. Vertex property theorem;

Examples of geometric expressions with equal vertex angles:

∠∠AOC =∠DOB

∴ ……………

8. Definition of two mutually perpendicular straight lines:

Two straight lines intersect into four angles, one of which is a right angle. These two straight lines are perpendicular to each other.

Example of geometric expression:

(1) ∵AB and CD are perpendicular to each other.

∴∠COB=90

(2)∫∠COB = 90

Ab and CD are perpendicular to each other.

9. Three lines parallel theorem:

Both straight lines are parallel to the third straight line, and so are the two straight lines (as shown in the figure).

Example of geometric expression:

∫AB∨EF

Also ∵CD∨EF

∴AB∥CD

10. parallel line judgment theorem;

Two straight lines are cut by a third straight line:

(1) If the isosceles angles are equal, then two straight lines are parallel; (pictured)

(2) If the internal dislocation angles are equal, two straight lines are parallel; (pictured)

(3) If the internal angles on the same side are complementary, the two straight lines are parallel (as shown in the figure).

Example of geometric expression:

( 1)√∠GEB =∠EFD

∴ab∑CD

(2)AEF =∠DFE

∴ab∑CD

(3) ∵∠BEF+∠DFE= 180

∴ab∑CD

1 1. The property theorem of parallel lines;

(1) Two parallel lines are cut by a third straight line and have the same angle; (pictured)

(2) Two parallel lines are cut by a third line, and the internal dislocation angles are equal; (pictured)

(3) Two parallel lines are cut by a third straight line, which are complementary to the inner angle of the side (as shown in the figure).

Example of geometric expression:

( 1)∵AB∨CD

∴∠GEB=∠EFD

(2)∵AB∨CD

∴∠AEF=∠DFE

(3)∵AB∨CD

∴∠BEF+∠DFE= 180

B-level concept of geometry: (requires understanding, speaking and using, mainly used for filling in the blanks and multiple-choice questions)

A basic concept:

Straight line, ray, line segment, angle, right angle, flat angle, fillet, acute angle, obtuse angle, complementary angle, adjacent complementary angle, distance between two points, intersection line, parallel line, vertical line segment, vertical foot, diagonal line, extension line and anti-extension line, congruent angle, internal angle, distance from point to line and distance between parallel lines.

Two theorems:

1. axiom of straight line: there is only one straight line after two points.

2. Axiom of line segment: The line segment between two points is the shortest.

3. Theorem about vertical line:

(1) There is one and only one straight line perpendicular to the known straight line;

(2) Of all the line segments connecting a point outside and a point on the line, the vertical line segment is the shortest.

4. Parallelism axiom: After passing a point outside a straight line, there is one and only one straight line parallel to this straight line.

Three formulas:

Right angle = 90, straight angle =180, fillet = 360, 1 = 60',1'= 60 ".

Four common sense:

The definition of 1. is bidirectional, but the theorem is not.

2. Straight lines cannot be extended; Ray can not extend forward, but can extend backward; Line segments can extend in two directions.

3. The proposition can be written in the form of "If ……………………………………………………………………………………………………".

4. Geometric drawing should draw general graphics, so as not to attach unnecessary conditions to the topic and cause misunderstanding.

5. When calculating rays, line segments and angles, they should be calculated in sequence or classification.

6. Geometric argumentation can be analyzed by analytical synthesis, equation analysis, substitution analysis and graphic observation.

7. Direction angle:

8. Scale: In the scale of 1:m, 1 represents the distance on the map and m represents the actual distance. If it is 1 cm on the map, it represents the actual distance of m cm.

9. The proof of geometric problems should use the "demonstration method", and the demonstration requires standardization, rigor and justification; Proof is based on learned definitions, axioms, theorems and inferences.