1. Among the following categories, there are _ _ _ _ _ _ _ _ _ _ _. ()

① ② ③ ④

⑤(x unknown)

A.2 B.3 C.4 D.5

Answer: b

2.(20 10 Zhejiang simulation, 15) The solution of the fractional equation is x = _ _ _ _ _ _ _ _ _ _ _ _ _ _

Answer: 1

3. If the fractional equation has an augmented root, the augmented root is _ _ _ _ _ _ _ _ _ _ _ _, and m = _ _ _ _ _ _ _ _ _.

Analysis: Multiply both sides of the equation by (x+3) to get x+2 = m, and solve this equation to get x=m-2. Because the fractional equation has an additional root, the additional root is x=-3. So -3=m-2, and the solution is m=- 1 So the additional root is x=-3.

Answer: x=-3-1

4. Solve the equation:

Solution: multiply both sides of the equation by x-3, and x-2=2(x-3)+ 1. Solve this equation, x=3.

Test: when x=3, x-3=3-3=0, so x=3 is the root of the original equation, and the original equation has no solution.

10 minute training (intensive training, which can be used in class)

1.Students from Class A and Class B take part in afforestation. It is known that Class A plants 5 more trees every day than Class B. The number of days for Class A to plant 80 trees is equal to the number of days for Class B to plant 70 trees. If Class A plants X trees every day, the equation listed according to the meaning of the question is ().

A.B. C. D。

Analysis: the equivalent relationship is: the number of days required for 80 trees of A species = the number of days required for 70 trees of B species. If class a plants x trees every day, the equation listed according to the meaning of the question is.

Answer: d

2. When method of substitution is used to solve equation () 2-+3x-6=0, if it is set, the original equation will be transformed into an equation about Y _ _ _ _ _ _ _ _ _ _ _ _

Analysis: 1. Deformation of the original equation: () 2+3()+6=0. After substitution, this equation is y2+3y-6=0.

Answer: y2+3y-6=0.

3. In order to control the sewage, a city needs to lay a sewage drainage pipeline with a total length of 3 000 m. In order to minimize the impact of construction on urban traffic, in actual construction, the daily efficiency is increased by 25% compared with the original plan. Therefore, the task was completed 30 days ahead of schedule. How long is the pipeline actually laid every day?

(1) If the pipeline x m is laid every day as originally planned, the equation listed is _ _ _ _ _ _ _ _ _ _ _ _.

(2) The meaning of the question is the same as above, and the question is changed to: How many days does it take to actually lay the pipeline?

Assuming that it takes X days to actually lay the pipeline, the equation listed is _ _ _ _ _ _ _ _ _ _ _ _.

Analysis: This question is changeable. (1) According to the equivalence relation of completing the task 30 days ahead of schedule, the equation can be listed: Suppose the original plan is to lay the pipeline x meters a day, but the actual laying of the pipeline is (1+25%)x meters a day, and according to the meaning of the question, the score is =30.

(2) According to the equivalence relation that the actual construction time efficiency is 25% higher than the original plan, the equation can be listed: Assuming that the actual pipeline laying takes x days, the original planned time is (x+30) days, and according to the meaning of the question, it needs X (1+25%).

Answer: (1)=30

(2)×( 1+25%)

4. When solving the equation, Xiao Liang's solution is as follows:

Solution: Multiply both sides of the equation by x-3 to get 2-x=- 1-2(x-3). Solving this equation gives x=3.

Do you think x=3 is the root of the original equation?

Solution: According to the steps of solving the fractional equation, the above solution does not test the root. When x=3 is substituted into the original equation and the denominator is zero, then x=3 is the root of the original equation and the original equation has no solution.

5. Solve the fractional equation.

Solution: First, find the common denominator (x+3)(x-3) with the simplest denominator, and use it to multiply the two sides of the equation, remove the denominator, and convert the fractional equation into an integral equation before solving.

Multiply both sides by (x+3)(x-3)

3(x+3)-(x-3)= 18，

3x-x= 18-3-9，

2x=6，

x=3。

Test: substitute x=3 into the original equation,

The denominator on the left (x-3)=3-3=0,

∴x=3 is the root of the original equation.

∴ The original equation has no solution.

6. Solve the equation:

Solution:

5(x+ 1)=3(x- 1)，

5x+5=3x-3，

2x=-8，

x=-4。

Test: substitute x=-4 into the original equation,

Left = right =- 1, so x=-4 is the root of the original equation.

7. What is the value of k, and the equation will generate roots?

Solution: This example is the same as the fractional equation, except that there is a undetermined coefficient K, and the value of K determines the value of the unknown quantity X, so X can be expressed by the algebraic expression of K, and a new equation can be established to solve it by combining the increase of roots with the simplest common denominator x-3=0.

If the denominator is removed, x-4(x-3)=k,

∴x=.

When x=3, the equation will generate roots.

∴=3.∴k=3.

30-minute training (consolidation training can be used after class)

1. The candle forms a real image through a convex lens. The object distance u, image distance v and focal length f of the convex lens satisfy the relationship. If u= 12 cm and f=3 cm, the value of v is ().

8 cm long, 6 cm wide, 4 cm high and 2 cm wide.

Analysis: substitute u= 12 and f=3 into the original equation.

Answer: c

2. If the equation has an increasing root, then its increasing root is ()

A.0b.1C.-1 D.1and-1

Analysis: According to the meaning of increasing root, the root with denominator of 0 is the increasing root of the original equation. Therefore, let (x+ 1)(x- 1)=0 and the solution is x=- 1 or x= 1.

Answer: d

3. In the following equation, there is no solution ().

A.B.

C.D.

Analysis: If the denominator is removed to solve the equation, X- 1 = X+ 1 and-1 = 1 appear in D, so D has no solution.

Answer: d

4.(20 10 Nantong Jiangsu simulation, 17) Solve the equation with method of substitution. If set, you can get the whole equation about y: _ _ _ _ _ _ _ _ _ _.

Analysis: The original equation is transformed into 2×=4.

Let =y, the original equation can be transformed into 2y+=4.

Arrange 2y2-4y+ 1=0.

Answer: 2y2-4y+ 1=0.

5. There are two experimental fields with the same area. The first one used the original variety, and the second one used the new variety, and harvested 9 000 kg and 15 000 kg wheat respectively. As we all know, the yield per hectare of the first experimental field is 3 000 kilograms less than that of the second experimental field, so we should calculate the yield per hectare of these two experimental fields separately.

Can you find out all the equivalence relations in this problem?

If the yield per hectare of the first experimental field is X kilograms, then the yield per hectare of the second experimental field is _ _ _ _ _ _ _. According to the meaning of the question, the equation _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

Analysis: Equivalence relations include:

Yield per hectare of the first experimental field +3 000 kg = yield per hectare of the second experimental field.

Yield per hectare =,

The area of the first experimental field = the area of the second experimental field.

The yield per hectare of the second experimental field is (x+3000) kg;

The equation is.

A: (x+3 000)

6. There are two expressways from A to B: one is an ordinary expressway with a total length of 600 kilometers and the other is an expressway with a total length of 480 kilometers. The average speed of buses traveling on expressways is 45 kilometers faster than that on ordinary highways, and the time required from expressways A to B is half that from ordinary highways A to B. Find the time required for the bus to travel from highway A to highway B.

What are the equivalence relations in this question?

If it takes X hours for a bus to take the expressway from A to B, then it takes _ _ _ _ _ _ _ _ hours to take the ordinary highway from A to B. According to the meaning of the question, the equation can be obtained as follows: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

Analysis: Equivalence relations include:

600 km = the average speed of passenger cars on ordinary roads × the time for passenger cars to travel from A to B on ordinary roads.

480 km = the average speed of the bus on the expressway × the time for the bus to travel from place A to place B on the expressway, the average speed of the bus on the expressway-the average speed of the bus on the ordinary highway =45 km/h,

Time from a to b on expressway =× time from a to b on ordinary highway.

Answer: 2x =45

7. Solve the equation.

Solution: The original equation can be transformed into ()+() = ()+().

That is to say,

The left side and the right side are separated,

Thereby obtaining (x-9)(x-8)=(x-6)(x-5),

The solution is x=7.

X=7 is the root of the original equation.

∴x=7.

8. A class organizes students to visit the Science and Technology Museum. In order to support the popular science activities carried out by the school, the Science and Technology Museum decided to charge students a one-time fee according to the minimum standard. The whole class was counted as 200 yuan, and 10 students were unable to participate in the activity for some reason. As a result, each student spent 1 yuan more than originally planned. How many students are planned to take part in this class?

Solution: Suppose there are X students participating in the activity.

Then = 1,

The solution is x 1=50, x2=-40.

After testing, x=50 is the root of the original equation, and x=-40 is not the meaning of the question, so it is abandoned.

A: It was originally planned that 50 people would participate in this activity.

9. Can you try to find the solution of this equation?

Solution: Multiply both sides of the equation by x(x+3 000).

9 000(x+3 000)= 15 000x。

Solve this equation and you get x=4 500.

10. In response to the call to hold a "green Olympics", Class 2, Grade 3 of a middle school plans to organize some students to plant trees voluntarily 180 trees. Due to the high enthusiasm of the students, the number of people actually participating in tree planting activities has increased by 50% compared with the original plan. As a result, two trees were planted less than originally planned. How many people actually took part in the tree planting activities?

Solution: Suppose that X people originally planned to participate in tree planting activities, but actually 1.5x people participated in tree planting activities.

Meaning from the question =2.

If the denominator is removed, the result is 3x = 90 and x = 30.

X=30 is the solution of the original equation.

1.5x= 1.5×30=45。

In fact, 45 people took part in the tree planting activities.