1. Examination scope: Chapter 1: Basic knowledge of algebra, Chapter 2: rational numbers, Chapter 3: addition and subtraction of algebraic expressions, and Chapter 4: linear equations of one variable.

(1) Concepts: positive number, negative number, non-negative number, integer, fraction, rational number, number axis, reciprocal, absolute value, algebraic sum, reciprocal, power, base, exponent, significant number, algebra, value of algebraic sum, monomial, polynomial, coefficient of monomial and simplex.

(2) Properties and operating rules

① Reduction of Criminal Law, Combination Law and Distribution Law;

② Nature of absolute value: |a|≥0, (A is rational number);

③ Rational number is relatively large;

④ Properties of the equation;

⑤ The principle of the same solution of the equation;

(3) Methods and rules: ① addition, subtraction, multiplication and division of rational numbers; ② multiplication of multiple numbers; ③ sign rule of multiplication results; ④ operation sequence rule; ⑤ scientific notation; ⑤ finding the square sum cube of a number with a calculator or a look-up table; ⑦ approximation and rounding; ⑧ merging similar items; ⑨ removing brackets and adding brackets.

Second, the selection of examples: The following are some problems that students often make or are not clear about in the learning process.

Example 1. What is a negative reciprocal?

A: Negative reciprocal is a common saying. Two numbers whose product is 1 are reciprocal, so two numbers whose product is-1 are reciprocal.

For example, if a×b= 1, then a and b are reciprocal, but if a×b=- 1, then a and b are reciprocal.

Example 2. Find the value of-1+3-5+7-9+-+1991-1993.

Solution: The original formula = (-1+3)+(-5+7)+(-9+11)+(-13+15)+-(-650)

= 2+2+2++2- 1993 (498 2)

=2×498- 1993

=-997

Note: The last of each pair of two numbers is exactly a multiple of 4 minus 1,1991= 4× 498-1.

Example 3. It is known that mx3+3nxy2+2x3-xy2+y does not contain cubic terms about X and Y, so find the value of 2m+3n.

Solution: mx3+3n xy2+2x3-xy2+y = (m+2) x3+(3n-1) xy2+y,

Because there is no cubic term, m+2=0, 3n- 1=0, so m=-2, n=,

So 2m+3n=2×(-2)+3× =-3.

Example 4. Among the following equations, the one that is not linear is ().

(A) 2x=0 (B) 5x+ 1=3x-8

(C) = + 1 (D) = + 1

Answer: (d) No, according to the definition of linear equation with one variable, it should be simplified before judging. D is simplified to 0x=, which does not conform to the definition of linear equation (a≠0), so it is not a linear equation. (See the definition of linear equation on page 202 of the textbook).

Example 5. Is (x2-x)+ 1 = x2 a linear equation?

A: This equation is a linear equation (ditto).

Because (X2-X)+ 1= X2.

Simplification: X2- X+ 1= X2.

- X+ 1=0

So it is a one-dimensional linear equation.

Example 6. Is the equation a linear equation?

A: No. But it can be transformed into two different linear equations =7 and =-7. So the equation has two solutions, which can be written as x 1=28 and x2=-28.

Note: A linear equation has only one solution.

Example 7. The bus is 200 meters long and the truck is 280 meters long. It takes 18 seconds to drive parallel and in the opposite direction from the front to the rear. The speed ratio of bus to truck is 5: 3. How many meters did two cars walk per second?

Analysis: From meeting to leaving the rear of the car, the distance traveled by the two cars * * * = the length of the bus+the length of the truck.

Solution: If the speed of the bus is 5x m/s and the speed of the truck is 3x m/s, this is derived from the meaning and equation of the problem:

18(5x+3x)=200+280

Solution: x=,

∴ 5x=，3x= 10 .

A: (omitted).

Example 8. A boat dragged the sampan upstream, and the rope broke the sampan. When the people on board found it, they immediately turned around and chased it. Five minutes later, they asked how long the towing rope had been broken, and the people on board found that the sampan was lost.

Analysis: at the beginning of this problem, the ship and the sampan can be regarded as running in opposite directions, and later they can be regarded as chasing problems.

Solution: let's assume that the speed of the ship in still water is one meter per minute, and the current speed is b meters per minute. After another x minutes, I found that the rope was broken, and the equation had to be:

(a-b)x+bx=5(a+b)-5b

(a-b+b)x=5(a+b-b)

ax=5a

x=5

A: (omitted).

Example 9. In an intelligence contest, it is stipulated that to give the bottom score of 50 points, you must answer 10 questions, give one of 10 points, and subtract 5 points for a wrong question. Xiaoming got 90 points in the exam. How many questions are right and how many are wrong?

Solution: If he is right on X, he is wrong on (10-x). Column equation:

50+ 10x-5( 10-x)=90

Solution: x = 6, 10-x = 4,

A: Yes, 6 lanes, wrong 4 lanes.

Example 10. A person climbs a mountain, walking 2 kilometers per hour when going up and 3 kilometers per hour when going down. Try to find the average speed.

Analysis: Many people mistakenly think that the average speed is =, which is wrong.

Solution: Let the distance up the mountain be X kilometers and the average speed be V kilometers per hour. Column equation:

A: (omitted).

Final simulation test (100 minute) (full mark 100 minute)

1. Fill in the blanks: (2 points for each small question, * * * 20 points)

(1) |-The reciprocal of | is _ _ _ _, and the absolute value is equal to _ _ _ _.

(2) The integer whose absolute value is not less than 2 and not more than 5 is _ _ _ _ _ _ _.

(3) The number of m divisible by 3 is expressed as _ _ _ _ _.

(4) The original price of a commodity is one yuan, and the price after 15% discount is _ _ _ _ _ _.

(5) The coefficient of single item-is _ _ _ _ _ _, and the degree is _ _ _ _ _.

(6) The polynomial -2x2y2-4xy+3x3+7 is a polynomial of degree _ _ _, and it is _ _ _ _ _ _ _ _ _ in descending order of X.

(7) According to the rounding method, two significant figures are reserved, and scientific notation indicates that 7984 should be _ _ _ _ _ _.

(8)-x6ym and 3x2ny2 are similar items, then m=_____, and n=_____.

(9) When a=_____, x=4 is the solution of equation a(x-2)=a+3x.

(10) The positions of points representing rational numbers A, B and C on the number axis are as follows: Simplify | C-B |+| A-B |-| A | = _ _ _ _ _ _ _ _ _ _.

Second, multiple-choice questions: (3 points for each question, ***30 points) Only one of the four options given in each question meets the requirements of the topic. Please fill in the letter code before the correct option in brackets.

(1) The correct one of the following calculations is ().

(A) -5-5=0 (B) 0-(-2)=-2

(C) 2-5=3 (D) (-4)-0=-4

(2) The error in the following operations is ()

(A) -32=9 (B) 0÷(-3)=0

(C)(- 1)4 = 1(D)(-3)÷(-)= 9

(3) The following comparison result is correct ()

(A) 3 >|-3 |(B)-7 & gt； five

(C)-0. 1 & gt； 0.0 1(D)-& lt； -

(4) the following judgment is correct ()

(1) Specify the origin, and the straight line with the length as the unit is called the number axis.

(B) -24 is pronounced as the fourth power of negative two.

(C) a is a rational number, then -a is a negative number.

(D) m-2n and 2n-m are reciprocal.

(5) The following calculation is correct ()

(A) 2x+3y=5xy (B) ab-ba=0

(C) 3x2-x2=3 (D) 4x2y-6xy2=-2xy2

(6) It is known that m2=(- )2, so the value of m is equal to ().

(a)-(b)-(c)- or (d)

(7) The following statement is correct in the process of removing the bracket ()

(A) 3(a-b)=3a-3b (B)a+(b-c)=a+b+c

(C)a-(b-C)= a-b-C(D)-2(a+3b)=-2a-3b

(8) The following four processes of solving equations are correct ()

(a) For transposed items, 3x-2 = x+3x+x=3-2.

(B) =- 1 Divide by the denominator to get 2(x+3)+3x-2=- 1.

(c) 2 (x+1)-3 (2x-1) = 22x+2-6x+3 without brackets = 2.

(D)-x=2 Transform the unknown coefficient into 1, and x=-

(9) It is known that the equation (a+2)x|a+ 1|+5=0 is a linear equation about x, so the value of a is ().

(A) -2 (B)-1 (C) 0 (D) -2 or 0

(10) A city collects monthly gas charges according to the following provisions. Gas consumption does not exceed 60 cubic meters, according to the 0.8 yuan per cubic meter; If it exceeds 60 cubic meters, the excess will be charged per cubic meter 1.5 yuan. It is known that user A's average monthly gas fee per cubic meter is 0.9 yuan, so user A should pay this month's gas fee ().

60 yuan (B) 63 yuan (C) 70 yuan (D) 130 yuan.

Third, calculate the following questions. (3 points for each small question, *** 12 points)

( 1) - 18- 14÷(-7)+4×(-3)

(2) [(- 18)+ 127.3+(- 182)]+2.7

(3) -3 -4 ÷(-2)3+(- 1)2×

(4) (-0.3)×( )×(-40) (using simple method)

Fourth, answer questions. (4 points for each small question, ***8 points)

(1) It is known that m2-mn=2 1 and mn-n2= 12. Find the values of algebra m2+n2-2mn and m2-n2.

(2) Simplified evaluation: 5x-{2y-3x+[5x-2(y-2x)+3y]}, where x=- and y=-

5. Solve the following one-dimensional linear equation. (5 points for each small question, *** 15 points)

( 1) + 1=x

(2) { [ ( -3)-3]-3}=0

(3) -6.5= -7.5

Sixth, use equations to solve application problems. (5 points for each small question, *** 15 points)

(1) A group of students went camping for military training outside the school. They marched at a speed of 5 kilometers per hour. When they left 18 minutes, the school had to send an urgent notice to the captain. The correspondent started from the school and rode his bike to catch up with the students at a speed of 14 km/h. How long did it take the correspondent to catch up with the students?

(2) The gross output value of a factory in the previous year was 250,000 yuan more than the total expenditure. Last year, the total output value and total expenditure were twice that of the previous year. This year, the total output value increased by 65,438+05%, and the total expenditure decreased by 65,438+00%. It is understood that the total output value this year is 950,000 yuan more than the total expenditure. What is the total output value and total expenditure this year?

(3) Teachers take students to travel. A travel agency said, "If the teacher buys a full ticket, the rest of the students can enjoy a half-price discount." B Travel Agency said, "Including the teacher, the full fare is 60% off." If the whole fare is 250 yuan, which travel agency is more favorable?

Reference answer:

First, fill in the blanks:

( 1)-(2)2，3，4，5 (3) 3m

(4)0.75 a(5)-3

(6)4.3x 3-2x2y 2-4xy+7(7)8.0× 103

(8) 2，3 (9) 12 ( 10) c

Second, multiple-choice questions:

( 1) D (2) A (3) D (4) D (5) B (6) C

(7) A (8) C (9) C ( 10) B

Third, the calculation questions:

( 1) -28 (2) -70 (3) -2

(4) solution: the original formula =(-0.3)×(-40)× ()

= 12×( )

= 1 1- 14+9=6

Fourth, answer questions:

(1) solution: ∫m2-Mn = 2 1, mn-n2= 12.

∴m2+N2-2mn = m2-Mn-Mn+N2 =(m2-Mn)-(Mn-N2)= 2 1- 12 = 9

m2-N2 = m2-Mn+Mn-N2 =(m2-Mn)+(Mn-N2)= 2 1+ 12 = 33

(2) Solution: The original formula = 5x-{2y-3x+5x-2y+4x+3y} = 5x-3y-6x =-x-3y.

When x=- and y=-, the original formula =-(- )-3×(- )=+= 1.

Five, solve the equation:

(1) solution:+1=x

3 (x-2)+5 (2x-1)+15 =15x after the denominator is removed.

3x-6+ 10x-5+ 15 = 15x

2x=4

∴ x=2

(2) Solution: {[(-3)-3]-3}=0

[ ( -3)-3]-3=0

( -3)-3=6

( -3)=9

-3= 18

=2 1

∴ x=42

(3) Solution: -6.5= -7.5

400-600 x+ 1 = 1- 100 x

500x=400

∴ x=

Six, column equation to solve application problems:

(1) Solution: Set the correspondent to catch up with the students for x hours.

From the meaning of the question: 14x=5× +5x.

9x=

∴ x=

A: The correspondent caught up with the students in hours (that is, 10 minutes).

(2) Solution: If the total output value of the previous year was X million yuan, the total output value of this year is 2x( 1+ 15%) million yuan, and the total expenditure is [2x( 1+ 15%)-95] million yuan.

From the meaning of the question, it is 2x (1+15%)-95 = 2 (x-25) (1-kloc-0/0%).

X= 100 when solving this equation.

∴ 2x (1+15%) = 200x115% = 230 (ten thousand yuan)

230-95= 135 (ten thousand yuan)

A: The total output value of the factory this year is 2.3 million yuan, and the total expenditure is 6.5438+0.35 million yuan.

(3) Solution: Suppose there are X students, then

A travel agency fee-B travel agency fee

=(250+250×0.5x)-250×0.6(x+ 1)

= 250+ 125 x- 150 x- 150

= 100-25x

When 100-25x >: 0, x

When 100-25x=0 and x=4, the two travel agencies are the same.

When 100-25x

A: when the number of students is less than 4, travel agency b is more favorable; When there are four students, the two travel agencies are the same; When the number of students is more than 4, the travel agency is more favorable.