Please help with some junior high school math questions!

Junior high school mathematics basic knowledge test questions

School name score

1. Fill in the blanks (this question has 30 questions, each question is worth 2 points, full score 60 points)

1. Sum and sum are collectively called real numbers.

2. The solution of the equation - =1 is .

3. The solution set of the inequality group is .

4. There are 100 five-cent and two-cent coins, worth 3 yuan and 20 cents. If there are x five-cent coins and y two-cent coins, we can get a system of equations.

5. Calculation: 28x6y2÷7x3y2＝．

6. Factorization: x3+x2-y3-y2＝.

7. When x, the fraction is meaningful; and when x, its value is zero.

8. Calculation: + =; (x2-y2)÷ =.

9. Expressed in scientific notation: -0.00002008＝; 121900000＝.

10. The square root of is ; the cube root of - is .

11. Calculation: - =; (3+2) 2=.

12. Rationalization of the denominator: ＝; ＝.

13. From a rectangular iron piece 8cm long and 6cm wide, cut off a small square with equal sides at each of the four corners to make a rectangular box without a lid, so that its base area is 24 cm2． If the side length of the small square is x cm, the equation can be obtained as .

14. If there are two unequal real roots in the x equation 2x2-4x+k=0, then the value range of k is .

15. If x1 and x2 are the two roots of the equation 2x2+6x-1=0, then + = .

16. The quadratic equation of one variable with +1 and -1 as roots is .

17. Factoring within the range of real numbers: 3x2-4x-1=.

18. The solution of the equation x+ =5 is ．

19. It is known that the proportional function y=kx, and when x=5, y=7, then when x=10, y=.

20. When k, if the inverse proportional function y= is in the quadrant where its graph is located, the function value increases as x decreases.

21. In the Cartesian coordinate system, the analytical formula of the straight line passing through the points (-2, 1) and (1, -5) is ．

22. If k<0, b>0, then the graph of the linear function y=kx+b passes through the first quadrant.

23. If the perimeter of an isosceles triangle is 24cm, then the functional relationship between the waist length y (cm) and the base length x (cm) is .

24. The opening of the graph of the quadratic function y=-2x2+4 x-3 is in the direction of ; the vertex is .

25. The analytical formula of the parabola passing through the points (1, 3), (-1, -7), (-2, -6) is ．

26. After moving the parabola y=-3(x-1)2+7 to the right by 3 units and down by 4 units, the analytical formula of the parabola obtained is .

27. Among the students in a certain class at Liuying Middle School, there are 18 students who are 14 years old, 16 students who are 15 years old, and 6 students who are 16 years old. The average age of students in this class is years.

28. When a set of data has 8 numbers arranged from small to large, the median of this set of data is .

29. A set of data has 80 numbers, the largest number is 168, and the smallest number is 122. If the group interval in the frequency distribution histogram is 5, this set of data can be divided into groups.

30. The standard deviation of samples 29, 23, 30, 27, and 31 is ．

2. Fill in the blanks (this question has 30 questions, each question is worth 2 points, the full score is 60 points)

31. If two parallel lines are connected by a third straight line, Intercept, then they are equal and complementary.

32. The hypothesis of the proposition "Two straight lines are parallel and interior angles on the same side are complementary" is ,

The conclusion is .

33. If the lengths of the three sides of the triangle are 6, 11, and m respectively, then the value range of m is .

34. If a polygon

The sum of the interior angles is 2520°, then this polygon is a polygon.

35. , , of an isosceles triangle overlap each other.

36. In △ABC, if ∠A=80° and ∠B=50°, then △ABC is a triangle.

37. In Rt△ABC, ∠C=90° and ∠A=60°. If AC=5cm, then AB=cm.

38. In Rt△ABC, ∠C=90°, if AC=3cm, BC=4cm, then the height CD of side AB= cm.

39. If the difference between two adjacent angles of a parallelogram is 30°, then the larger interior angle of the parallelogram is (degrees).

40. A quadrilateral with two opposite sides is a parallelogram.

41. In the rhombus ABCD, if there is an interior angle of 120° and the shorter diagonal is 12cm long, then the perimeter of the rhombus is cm.

42. A parallelogram with two diagonals is a square.

43. In the trapezoid ABCD, AD‖BC, if AB=DC, then the equal base angles are ．

44. The figure obtained by connecting the midpoints of the four sides of the rhombus in sequence is a shape.

45. In △ABC, points D and E are on the sides AB and AC respectively. If DE‖BC, AD=5, AB=9, EC=3, then AC=.

46. In △ABC, points D and E are on sides AB and AC respectively, AD=2 cm, DB=4cm, AE=3cm, EC=1 cm. Because and, so △ABC ∽△ADE．

47. The three midlines AD, BE, and CF of △ABC intersect at point G. If the area of ??△AEG is 12 square centimeters, then the area of ??△ABC is 12 square centimeters.

48. Change a triangle into a triangle similar to it. If the side length is expanded to 10 times of the original, then the area will be expanded to 10 times of the original.

49. If ∠A is an acute angle, tgA=, then ctgA=.

50. Calculation: sin30°=; tg60°=.

51. In Rt△ABC, ∠C=90°. If sinA=, then ∠B= (degree).

52. If an airplane looks down at a target on the ground at an altitude of 5,000 meters above the ground, and the depression angle is 30°, then the distance between the airplane and the target is meters.

53. The slope of the slope is 1:4, the horizontal width of the slope is 20m, then the vertical height of the slope is m.

54. In a circle with a radius of 10cm, the length of the arc subtended by the central angle of 20° is cm.

55. If the radii of the two circles are 9cm and 4cm respectively, and the distance between the centers of the circles is 5cm, then the positional relationship between the two circles is ．

56. If straight line AB passes through point C on ⊙O, and OC⊥AB, then straight line AB is of ⊙O.

57. In △ABC, if AB=9cm, BC=4cm, CA=7cm, and its inscribed circle cuts AB at point D, then AD= cm.

58. In Rt△ABC, ∠C=90°. If AC=5cm, BC=12cm, then the radius of the inscribed circle of △ABC is cm.

59. Two circles with radii of 5cm and 15cm are circumscribed. The length of the tangent line to the grandpa is cm, and the acute angle between the center line and the tangent line to the grandpa is (degree).

60. Any regular polygon is a symmetrical figure, and a regular polygon with an even number of sides is also a symmetrical figure.

Answer

1. 1. Rational numbers; irrational numbers. 2. y=3. 3. x≤-． 4. ． 5. 4x3． 6. (x-y) (x2+xy+y2+x+y). 7. ≠- ;=1. 8. ;(x+y)2. 9. -2.008×10-5; 1.219×108． 10. ±3;-． 11. ;29+12． 12. ;．． 13. (8-2x) (6-2x)=24 (or x2-7x+6=0). 14. k<2. 15. 6. 16. x2-2 x+1＝0. 17.

(x- ) (x- ). 18. x=3. 19, 14. 20.＞0． 21. y=-2x-3. 22. One, two, four. 23. y＝-x＋12, 0＜x＜12. 24. Down; (1, -1). 25. y=2x2+5x-4． 26. y=-3(x-4)2+3. 27, 14.7． 28. The average of the 4th and 5th numbers. 29, 10． 30. 2．

2. 31. Isotopic angles or interior angles; homoparallel interior angles. 32. Two straight lines are parallel; interior angles on the same side are complementary. 33. 5＜m＜17. 34, 16. 35. The bisector of the vertex angle; the midline on the bottom edge; the height on the bottom edge. 36. Isosceles. 37, 10. 38, 2.4． 39. 105°． 40. Parallel (or equal). 41, 48. 42. Vertical and equal. 43. ∠A＝∠D, ∠B＝∠C. 44. Moment． 45. 46. ??∠DAE＝∠CAB, =． 47, 72. 48, 100． 49. ． 50.; ． 51. 30°． 52. 10000. 53.5． 54. π． 55. Incision. 56. Tangent. 57. 6. 58.2． 59, 10; 30°． 60. Axis; center.

Basic Test of "Preliminary Knowledge of Algebra"

1 Fill-in-the-blank questions (20 points for this question, 4 points for each question):

1. The side length of the square is a cm. If each side of the square is reduced by 1 cm, the area of ??the square after the reduction is

cm2;

2. a, b, c represent three rational numbers, and the additive associative law expressed by a, b, c is ;

3. The difference between x and y 7 times is expressed as ;

4. When , the value of the algebraic expression is ;

5. The solution to the equation x-3 = 7 is .

Answer:

1. (a-1)2;

2. a+(b+c)=(a+b)+c;

3. x-7y;

4.1;

5.10.

Two multiple-choice questions (30 points for this question, 6 points for each question):

1. The following formulas are algebraic:……………………………………………………………… ( )

(A) S = πr (B) 5＞3 (C) 3x-2 (D) a＜b＋c

2. The number A is 2 greater than the number B. If the number B is y, the number A can be expressed as………………………………( )

(A) y+2 (B) y-2 ( C) 7y+2 (D) 7y-2

3. Among the following formulas, the equation is……………………………………………………………… ( )

(A) 2+5=7 (B) x+8 ( C) 5x+y=7 (D) ax+b

4. A three-digit number, the ones digit is a, the tens digit is b, and the hundreds digit is c. This three-digit number can be expressed as ( )

(A) abc (B) 100a＋10b＋c (C) 100abc (D) 100c＋10b＋a

5. The output value of a factory in January was a million yuan, and the production increased by 15% in February. The output value in February can be expressed as ( )

(A) (1+15%) × a million yuan (B) 15% × a Ten thousand yuan

Answer:

1 ． C; 2. A; 3. C;4. D;5． A．

3. Find the value of the following algebraic expression (10 points for this question, 5 points for each question):

1.2×x2+x-1 (where x = );

Solution: 2×x2＋x－1

＝

＝2× ＋ -1＝＋ -1＝0;

2. (in ).

Solution: = =．

Four (10 points for this question)

As shown in the picture, there is a largest circle in an isosceles trapezoid. The upper base of the trapezoid is 5cm, the lower base is 7cm, and the half of the circle is

The diameter is 3cm. Find the area of ??the shaded part in the figure.

Solution: It is known that the height of the trapezoid is 6cm, so the area S of the trapezoid is

＝×（a＋b）×h

＝×（5＋7 )×6

= 36 (cm2)．

The area of ??the circle is

(cm2).

So the area of ??the shaded part is

(cm2).

5. Solve the following equations (10 points for this question, 5 points for each question):

1.5x-8 = 2; 2. x+6 = 21．

Solution: 5x = 10, Solution: x = 15,

x = 2; x =15 =15 × =25.

Application problems of solving six series equations (20 points for this question, 10 points for each question):

1. A and B are practicing a race. If A lets B run 10 meters first, A can catch up with B in 5 seconds. If A runs 9 meters per second, what should be B's speed?

Solution: Assume that the speed of B is x meters per second, the equation can be written

(9-x)×5 = 10,

The solution is x = 7 (m/s)

2. It costs 2 yuan and 5 cents to buy three pencils and one ballpoint pen. If the ballpoint pen sells for 1 yuan and 60 cents, what is the price of the pencil?

Solution: Assume that the selling price of pencils is x yuan, and the equation can be written

3x+1.6 = 2.05,

The solution is x = 0.15 (yuan)

Basic test of "Quadratic Radicals"

(1) True or False Questions: (1 point for each question, ***5 points).

1. =2. ...( ) 2. It is a quadratic radical. ……………… ( )

3. ＝＝13－12＝1． ( )4． , , are quadratic radicals of the same type. ......( )

5. The rationalization factor of is . …………( ) Answer 1. √；2． ×；3． ×；4． √；5． ×.

(2) Fill-in-the-blank questions: (2 points for each question, ***20 points)

6. The condition for the equation =1-x to hold is _____________. The answer is x≤1．

7. When x____________, the quadratic radical makes sense. Hint: What are the conditions for a quadratic radical to be meaningful? a≥0． Answer ≥ ．

8. Compare size: -2______2-. Tips ∵ , ∴ , . Answer＜.

9． Calculation: equal to __________. Hint (3 )2-( )2=? Answer 2.

10. Calculation: ? =______________. Answer .

11. The positions of the corresponding points of the real numbers a and b on the number axis are as shown in the figure: a o b then 3a- =______________.

Tip: Find out from the number line what numbers a and b are? a<0, b>0. Are 3a-4b positive or negative?

3a-4b<0. Answers 6a-4b.

12． If + =0, then x=___________, y=_______________.

What do tips and mean? [The arithmetic square root of x-8 and y-2, the arithmetic square root must be non-negative,] What conclusion can you draw? [x-8=0, y-2=0. ] Answer 8, 2.

13. The rationalization factor of 3-2 is ____________.

Hint (3-2) (3+2)=-11. The answer is 3+2.

14. When ＜x＜1, - =______________.

Prompt x2-2x+1=( )2; -x+x2=( )2. [x-1; -x. ] When

15． If the simplest quadratic radicals and are quadratic radicals of the same kind, then a=__

___________,

b＝______________.

Hint What is the exponent of the radical of the quadratic radical? [3b-1=2. What is the relationship between ]a+2 and 4b-a? Are the two equations the same kind of quadratic roots? [a+2=4b-a. ]

Answer 1,1.

(3) Multiple choice questions: (each question is worth 3 points, maximum 15 points)

16. Among the following deformations, the correct one is... ( ) (A) (2 ) 2 = 2 × 3 = 6 (B) = -

Comments: This question examines the properties of quadratic radicals. Note that (B) is incorrect because =|- |=; (C) is incorrect because there is no formula =.

17. Among the following formulas, it must be true... ( ) (A) = a + b (B) = a2 + 1

Comment on this question to examine the conditions under which the properties of quadratic radicals are established. (A) is incorrect because a+b is not necessarily non-negative. To be true, (C) must be a≥1. To be true, (D) must be a≥0 and b>0.

18. If the formula - +1 is meaningful, the value range of x is……………………( )

(A)x≥ (B)x≤ (C)x= (D ) None of the above are correct

Hint To make the formula meaningful, it must

Answer C.

19. When a＜0, b＜0, convert it into the simplest quadratic root, and we get………………………………( )

(A) (B) - ( C)-(D)

Hint = =． Answer B．

Comment on the properties of this question: |a| and the denominator are rationalized. Note that the reason for the error in (A) is that the number is not considered when applying properties.

20． When a<0, the result of simplifying |2a- | is... ( ) (A) a (B) - a (C) 3a (D) - 3a

Prompt to simplify first, ∵ a<0, ∴ =-a. Then simplify |2a- |=|3a|. Answer D.

(4) Factoring within the range of real numbers: (4 points for each question, ***8 points)

21. 2x2-4; prompt to extract 2 first, and then use Square difference formula. Answer 2 (x+) (x-).

22． x4－2x2－3． Tips: First treat x2 as a whole, and use x2+px+q=(x+a)(x+b) where a+b=p, ab=q to decompose. Then use the square difference formula to decompose x2-3. Answer (x2+1) (x+) (x-).

(5) Calculation: (5 points for each question, ***20 points)

23. (-)-(-);

Tips: first convert each quadratic radical into the simplest quadratic radical, and then merge similar quadratic radicals. Answer .

24. (5 + - )÷;

Solution to the original formula = (20 +2 - )× =20 × +2 × - ×

＝22- × =22-2.

25. + -4 +2(-1)0; original solution=5 +2(-1)-4× +2×1

＝5 +2 -2-2 +2＝5.

26. (－＋2＋)÷．

Tips for this question: first convert division into multiplication, use the distributive law to multiply, and then simplify.

The original solution = (- ＋2 +)?

＝ ? - ＋2 ? ＋ ? ＝ - ＋2＋＝a2＋a- ＋2.

Comments on this question: It would be tedious to simplify the terms in the brackets first, then use the distributive law to multiply them and then simplify them.

(6) Evaluation: (6 points for each question, ***18 points)

27. It is known that a= , b= , find the value of -.

Tip to simplify the quadratic radical first and then substitute it into the evaluation.

Solve the original formula = = =．

When a= , b= , the original formula = =2

．

Comments: If the values ??of a and b are directly substituted into the calculation, the calculation process will be more complicated and calculation errors will easily occur.

28. Given that x= , find the value of x2-x+.

Tips for this question: x should be simplified first and then evaluated.

Solve ∵ x＝＝．

∴ x2－x＋＝( ＋2)2－( ＋2)＋＝5＋4 ＋4－－2＋＝7＋4．

Comments If we can notice that x-2= , so (x-2)2=5, we can also transform x2-x+ into the quadratic three about

x-2 Formula, the following solution is obtained:

∵ x2-x+ = (x-2)2+3 (x-2) +2+ = ( )2+3 +2+ =7+4.

Obviously the operation is convenient, but the requirements for the identity deformation of the formula are very high.

29. Given that + =0, find the value of (x+y)x.

Hint, are both arithmetic square roots, therefore, they are both non-negative numbers. What is the conclusion that the sum of two non-negative numbers is equal to 0?

Solve ∵ ≥0, ≥0,

And + =0,

∴ The solution is ∴ (x＋y)x＝(2＋1)2＝9.

(7) Answer the questions:

30. (7 points) It is known that the length of the hypotenuse of a right triangle is (2 +) cm and the length of the right angle side is (+2) cm. Find the area of ??this right triangle.

Tips for this question: What do you need to find the area of ??a right triangle? [Another right-angled side. ]How to request? [Using the Pythagorean Theorem. ]

The solution is in a right triangle, according to the Pythagorean theorem:

The length of the other right-angled side is: =3 (cm).

∴ The area of ??the right triangle is:

S＝×3×( )＝ (cm2)

Answer: The area of ??this right triangle is ( ) cm2 ．

31. (7 points) Given that |1-x|- =2x-5, find the value range of x.

Hint: From what is known, we get |1-x|-|x-4|=2x-5. When was this formula established? [1-x≤0 and x-4≤0. ]

The solution is known, the left side of the equation = |1-x|- =|1-x|-|x-4 and the right side = 2x-5.

Only when |1-x|=x-1 and |x-4|=4-x, the left side = the right side. At this time, the solution is 1≤x≤4. ∴ The value range of x is 1≤x≤4.

Basic Test of Quadratic Equations

(1) Fill-in-the-blank questions (2 points for each blank, ***26 points):

1. It is known that the linear equation of two variables = 0, and x is represented by an algebraic expression containing y, then x = _________;

When y=-2, x=___ ____. Tip: Taking y as a known number, solve for x. The answer is x=; x=.

2． Among the three sets of values ????(1), (2), and (3), _____ is the solution of the system of equations x-3y=9, ______ is the solution of the system of equations 2 x+y=4, and ______ is the solution of the system of equations. Prompt to substitute the three sets of values ??into the equation and the system of equations respectively for verification. Answers (1), (2); (1), (3); (1). Comment The solution to a system of equations must be the most identical solution to each equation in the system of equations.

3. It is known that , is the solution of the equation x+2 my+7=0, then m=_______. Tip: Substitute into the equation to find m. Answer- .

4． If the solution of the system of equations is , then a=__, b=_. Tip: Substitute into , the original system of equations is converted into a system of linear equations of two variables about a and b, and then solve it. The answer is a=-5, b=3.

5． It is known that the equation y=kx+b, when x=2, y=-2; when x=-, y=3, then k=____, b=____.

Tips: Substitute the corresponding values ??of x and y to get a system of linear equations of two variables about k and b.

The answer is k=-2, b=2. Comments It is a common method to solve the undetermined coefficients by establishing a system of equations.

6． If |3a＋4b－c|＋ (c－2 b)2=0, then

a∶b∶c=___________．

Hint: From the properties of non-negative numbers, we get 3 a+4 b-c=0, and c-2b=0. Then use the algebraic expression containing b to express a and c, so as to find the values ??of a, b, c. The answer is a=-b, c=2b; a:b:c=-2:3:6.

Comments: Using an algebraic expression of an unknown number to represent the remaining unknown numbers is a commonly used and effective method.

7. When m=_______, the equations x+2y=2, 2x+y=7, mx-y=0 have common solutions.

Tip to solve the system of equations first, and substitute the obtained values ??of x and y into the equation mx-y=0, or solve the system of equations

Answer, m=-. Comment "Common solution" is the basis for establishing a system of equations.

8. A three-digit number, if the number in the hundreds place is x, the number in the tens place is y, and the number in the ones place is twice the difference between the hundreds place and the tens place, then this three-digit number is _______________ ．

Tip to multiply the number in each digit by the corresponding number of digits and then sum.

The answer is 100 x+10 y+2 (x-y).

(2) Multiple choice questions (2 points for each question, maximum 16 points):

9. The following system of equations is known: (1), (2), (3), (4),

The number of equations belonging to the system of linear equations of two variables is……………… ……………………( )

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

Hint that the system of equations (2) contains three As for the unknown number, the degree of y in equation (3) is not 1, so (2) and (3) are not linear equations of two variables. Answer B．

10. It is known that 2 xb＋5y3a and -4 x2ay2-4b are similar terms, then the value of ba is……………………( )

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

Hints are defined by similar terms, get, solve, so ba=(-1)2=1. Answer C.

11. It is known that the solution of the system of equations is, then the values ??of m and n are... ( )

(A) (B) (C) (D)

The prompt will be substituted into the system of equations , we can solve the system of linear equations of two variables with respect to m and n. Answer D．

12． The solution of the system of linear equations in three variables is…………………………………………( )

(A) (B) (C) (D)

Tip to add both sides of the three equations separately to get x+y+z=6 or substitute the options into the equation set one by one to verify. From

x+y=1, we know that (B) and (D) are both wrong; and then use y+z=5, excluding (C), so (A) is correct. The former solution is called the direct method; the latter solution is called the inverse verification method. Answer A.

Comments: Because most of the multiple-choice questions in mathematics are single-choice questions - there is and only one correct answer, it has one more known condition than the general questions: there is and only one of the multiple-choice questions is correct. Therefore, in addition to the direct method, there are many special solutions to multiple-choice questions. As the study progresses, we will introduce them to the students one by one.

13. If the values ??of x and y of the solutions to the system of equations are equal, then the value of a is………………( )

(A)-4 (B) 4 (C) 2 (D) 1

Tips to substitute x=y into 4x+3y=14, solve to get x=y=2, and then substitute into the equation containing a. Answer C.

14. If the solution to the system of equations about x and y satisfies the equation 2x+3y=6, then the value of k is ( )

(A) - (B) (C) - (D) -

Tips: Treat k as a known constant, find the values ??of x and y, and then substitute the values ??of x and y into 2 x+3 y=6 to find k. Answer B.

15． If the equation y=kx+b, when x and y are opposite numbers to each other, b is 1 less than k, and x=, then the values ??of k and b are respectively......( )

(A) 2, 1 (B) , (C) - 2, 1 (D) , - Hint: From the known x=, y=-, the answer D can be obtained.

16. Students in a certain class are divided into groups to engage in activities. If there are 7 students in each group, there will be 4 students left; if there are 8 students in each group, there will be one group with 3 students missing. Assume that the class has

Hint: From the meaning of the question, we can get an equal relationship: (1) The number of students in group 7 = total number of students - 4; (2) The number of students in group 8 = total number of students + 3. Answer C.

(3) Solve the following system of equations (4 points for each question, ***20 points):

17. Tip: Use addition, subtraction and elimination to eliminate x first. Answer

18. Tips: First organize the equations into a system of equations with integer coefficients, and use addition and subtraction to eliminate x. Answer

19. Tip: From the first equation, we get x = y, and substitute it into the second equation; or from the first equation, assume x = 2 k, y = 5 k, and substitute it into another equation to find the k value. Answer

20. (a and b are non-zero constants)

Tip: Add the left and right sides of the two equations to get x+y=2a ①. Solve ① simultaneously with the two equations.

Answer

Comments: Superposition elimination is a common solution to systems of rotational equations with unknown number systems.

21.

Tip to connect the first equation with the other two equations, and use addition to eliminate y.

Answer

Comments Analyzing the composition characteristics of each unknown coefficient in each equation that makes up the system of equations is the key to choosing an appropriate method to solve the problem. Therefore, you must observe carefully before solving the problem. , can we find shortcuts to solve problems.

(4) Answer questions (6 points for each question, maximum 18 points):

22. It is known that the sum of x and y of the solution to the system of equations is 12, find the value of n.

Tip to solve the known system of equations, use the algebraic expression of n to express x and y, and then substitute x + y = 12.

Answer n=14.

23． Given that the system of equations has the same solution as , find the value of a2+2ab+b2.

Tip: Solve the system of equations first to find x and y, and then substitute them into the system of equations to find a and b.

Answer.

Comments When n systems of equations have the same solution, any two equations in the system of equations can be connected to form a new system of equations.

24. It is known that the values ??of the algebraic expression x2+ax+b when x=1 and x=-3 are 0 and 14 respectively. Find the value of the algebraic expression when x=3.

Hint: From the question, we can get the system of equations about a and b. Find a and b. Write this algebraic expression, and then find its value when x=3.

Answer 5.

Comments on this example: After using the undetermined coefficient method to find the values ??of a and b, you should write this algebraic expression, because it is a key step in the evaluation.

(5) Application problems of solving a system of equations (10 points for each question, ***20 points):

25. Last year, there were 80 more boys than girls in the first grade of a school. This year, the number of girls increased by 20% and the number of boys decreased by 25%. As a result, there were 30 more girls than boys. How many boys and girls were there in the first grade last year?

Tip: Suppose there are x boys and y girls in the first grade last year. We can get a system of equations

The answers are x=280 and y=200.

26. Places A and B are 20 kilometers apart. Two people, A and B, are traveling towards each other from places A and B at the same time. They meet on the way two hours later. Then A returns to place A, and B continues to move forward. When A returns to place A, B is still 2 kilometers away from place A. Find the speed of A and B.

From the meaning of the question, A walked for 2 hours before the encounter, and "when A returns to A, B is still 2 kilometers away from A", we can get another equal equation of the series. Relationship: A and B travel in the same direction for 2 hours, with a difference of 2 kilometers. Assume that the speeds of A and B are x kilometers/hour and y kilometers/hour respectively, then

The answer is that A’s speed is 5.5 kilometers/hour and B’s speed is 4.5 kilometers/hour. ．

Basic Test of "Fractions"

1 Fill-in-the-blank questions (2 points for each question, maximum 10 points):

1. Known v

=vat (a is not zero), then t= ;

2. The equation about x mx=a (the solution of m is;

3. The root of the equation is;

4. If -3 is the increasing root of the fractional equation, then a= ;

5. A car can travel x kilometers in a hour at the same speed.

Answer:

1 ．;2.;3． =2, express y with an algebraic expression containing y=2x-8 (D) y=2x-10

2. Among the following equations about x, the ones that are not fractional equations are……………………………… )

(A) (B)

3. A can complete a project alone in a hour, and B can do b alone. Hours to complete, the number of hours required for A and B to complete this work together is………………………………………………………………………… ( )

(A) a + b (B) (C) (D)

4. Solve the equation about x (m2-1) x=m2-m-2 (m2≠1) It should be expressed as.........( )

(A) x＝ (B) x＝

Answer: 1. D; 2. C; 4. B. 3. Solve the following equations (8 points for each question) ):

1．

, ,

After testing, =1 is the root of the original equation. After testing, =2 is the root of the original equation. Increase roots.

3. ;

Solution: Remove the denominator, we get,

Arrange the equations, we get

．

After verification, =2 is the root of the original equation.