Solution: the number of people in factory a is x, and the number of people in factory b is y.
There are two equal relationships in the problem: 1, and the number of people who arrive at a factory after drawing 9 people = the number of people who arrive at b factory.
The countable equation is: x-9=
2. The number of people who arrive at the factory after extracting 5 people =
The countable equation is:
(Financial distribution) Xiaohua bought stamps of 10 and 20 stamps of *** 16, which cost 2.50 yuan. How small are the 10 stamps and 20 stamps he bought? Solutions; Set * * * to buy X 10 stamps and Y 20 stamps.
Two equal relations in the problem:
1, 10 stamps +20 stamps = total number of stamps.
The countable equation is:
2. Total price 10 stamp+= total price of all stamps.
The countable equation is: 10X+ =
Xiaolan works in a toy factory. It takes 3 hours and 42 minutes to make 4 puppies and 7 cars, and 3 hours and 37 minutes to make 5 puppies and 6 cars. How long does it take to make 1 puppy and 1 car on average?
Two equal relations in the problem:
1, the time is four puppies +=3: 42.
The countable equation is:
2.+ Time to make six cars =3: 37.
The countable equation is:
(Travel problem) A and B are 6km apart and walk in the same direction. A can catch up with B within 3 hours; Walk in opposite directions and meet within 1 hour. What is their average speed? Solution: Let A walk X kilometers per hour and B walk Y kilometers per hour.
Two equal relations in the problem:
1, walking in the same direction: distance of a = distance of b+
The countable equation is:
2. Go in the opposite direction: the distance of A+=
The countable equation is:
A city has a population of 420 thousand. One year after the plan, the urban population will increase by 0.8% and the rural population will increase by 1. 1%, so the population of the whole city will increase by 1%. What is the current urban and rural population of this city?
Solution: At present, the urban population of this city is X million, and the rural population is Y million.
Two equal relations in the problem:
1, current urban population+= current total urban population.
The countable equation is:
2. The increased urban population next year+= the total population of the whole city next year.
The countable equation is: (1+0.8%) x+=
(Distribution problem) A kindergarten divides apples. If everyone has three apples, there are two left. If everyone has four apples, there is one less than 1. How many children are there in the kindergarten? Solution: There are X children in kindergarten and Y children in Pingguo.
Two equal relations in the problem: 1, the total number of flat fruits = 3 points per person+
The countable equation is:
2. Total number of apples =
The countable equation is:
(Concentration distribution) Prepare 45% saline 12kg, with 10% saline and 85% saline. How much do you need for each of these two kinds of brine?
Solution: Let the salt water with salt content of 10% be X kg and the salt water with salt content of 85% be Y kg. Two equal relations in the problem:
1, salt content 10% salt water+salt content in 85% salt water =
The countable equation is: 10%x+ =
2. Weight of salt water containing 10%+weight of salt water containing 85% =
The countable equation is: x+y=
(Financial appropriation) How much does it cost to mix candy sold in 4.2 yuan with candy sold in 3.4 yuan to make 200 kilograms of mixed candy sold in 3.6 yuan? Solution: Suppose that each kilogram of candy sold in 4.2 yuan is X kilograms, and each kilogram of candy sold in 3.4 yuan is Y kilograms.
Two equal relations in the problem:
1, with the total selling price of candy sold in 4.2 yuan per kilogram+=
The countable equation is:
2. The weight of candy is 4.2 yuan+= per kilogram.
The countable equation is:
(Geometric allocation problem) As shown in the figure, eight identical rectangles are used to form a large rectangle with a width of 48 cm. What is the length and width of each small rectangle? Solution: Let a small rectangle be x cm long and y cm wide.
Two equal relations in the problem:
1, length of small rectangle+= width of large rectangle.
The countable equation is:
2. Length of small rectangle =
The countable equation is:
A table consists of a table top and four legs. With 1 m3 of wood, you can make 50 tables or 300 legs. There are 5 cubic meters of wood. How to distribute wood to match the table top and legs?
Solution: Use
Two equal relationships in the title: 1, wood for desktop+=
The countable equation is:
2. Total number of all desktops: total number of all branches =
The countable equation is:
(Sum and difference multiplied by problem) A two-digit number, the number in the tenth place is 5 larger than the number in the first digit. If the number in the tenth place is exchanged with the number in the first place, the new two-digit number is 9, which is less than half of the original two-digit number. How to find this two-digit number?
Solution: Let one digit be X and ten digits be Y. Two equal relations in the problem:
1, unit number = -5
The countable equation is:
2. New two digits =
The countable equation is:
(Distribution and Transportation) A batch of goods is to be transported to a certain place, and the owner is going to rent two kinds of trucks from the automobile transportation company. It is known that these two kinds of trucks were rented in the past as shown in the left table. Now we rent five kinds of trucks of the company and six kinds of trucks of the company, and the goods are just delivered at one time. How many tons are these goods?
Solution: Suppose.
Two equal relations in the problem:
1, the first time: the weight of goods transported by truck A+=36.
The countable equation is:
2. The second time: the weight of goods transported by truck A+=26.
The countable equation is:
Detection of application problems of binary linear equations and re-exploration of practical problems
◆ Classification training of knowledge points
Knowledge point 1
1. There are 32 boys and girls in the class, and half of the girls are less than the total number of boys 10. If the number of boys is X and the number of girls is Y, the equation can be listed as follows.
2. The sum of numbers A and B is 10, and the difference is 2. If the number a is x and the number b is y, the equation can be listed as follows
3. If two solutions of the equation y=kx+b are known, then k= b=
The current annual output value of a factory is 6.5438+0.5 million yuan. If an additional investment of 654.38+0.000 yuan can increase the annual output value by 2,500 yuan, assuming an additional investment of X million yuan and a total output value of Y million yuan, then the equations satisfied by X and Y are as follows
5. The school used 250 yuan to buy 35 movie tickets * * * in which A ticket is for 8 yuan and B ticket is for 6 yuan. Let A ticket be X and B ticket be Y, and then list the equations. The solution of the equations is
6. A wooden stick is 8 meters long and divided into two sections, one of which is 1 meter longer than the other. In order to find the length of these two sections, suppose that one section is x meters and the other section is y meters, then the binary linear equation of the column is
7. If the circumference of a rectangle is 20 cm and the length ratio is 2 cm, then the length and width of the rectangle are both cm.
8. Athletes in a school are trained in groups. If there are 7 people in each group, the remaining 3 people; If there are 8 people in each group, there are 5 people missing; Let the number of athletes be x and the number of groups be y, then the equations are ().
9. When the speed of the ship is 40 km/h downstream and 26 km/h downstream, the speed of the ship in still water is
_ _ _ _ _ _ _, and the water flow speed is _ _ _ _.
10. A car starts from place A and goes east. On the way, it must pass a bridge. It takes the same time. If the speed is 60 kilometers per hour, you can cross the bridge for 2 kilometers. If the speed is 50 kilometers per hour, the distance to the bridge is only 3 kilometers, so the distance from A to the bridge is _ _ _ _ _ kilometers, which takes several hours. Think of the bridge as a point. )
1 1, the length of the rectangular lawn is twice the width, 10m, the perimeter is 132m, and the width and length are _ _ _ _ _.
12. A batch of books was distributed to a group of students, with 6 books each and 5 books each. The group * * * has _ _ _ students, and this batch of books * * * has _ _ _ _ _.
13. There are 246 students in a certain grade, including 3 boys less than girls and 2 times less than girls. How many boys and girls are there? Let the number of girls be x and the number of boys be y, then the system of equations _ _ _ _.
14, length of rope a and rope b * * * 17m. If the rope A is subtracted and the rope B is increased by 1m, and the lengths of the two ropes are equal, how many meters are the two ropes A and B respectively? If rope a is x(m) and rope b is y(m), equation () can be listed.
15. As we all know, the Yangtze River is 836 kilometers longer than the Yellow River, and the length of the Yellow River is six times that of the Yangtze River, that is, 1284 kilometers ... Let the lengths of the Yangtze River and the Yellow River be x(km) and y(km), respectively, then the equations can be listed.
16. There are 32 boys and girls in the class, and half of the girls are less than the total number of boys 10. If the number of boys is X and the number of girls is Y, the equation can be listed as follows.
17 and the sum of a and b is 10, with a difference of 2. If a is x and b is y, the equation can be listed as follows
18, k= b= if two solutions of the equation y=kx+b are known.
19. The current annual output value of a factory is10.5 million yuan. If every increase in investment 1000 yuan can increase the annual output value by 2500 yuan, assuming that the new investment is X million yuan and the total output value is Y million yuan, then the equation satisfied by X and Y is
On the 20th, the school used 250 yuan to buy 35 movie tickets * * *, including a ticket for each 8 yuan and a ticket for each 6 yuan. Let A ticket be X and B ticket be Y, then the equations are listed. The solution of the equations is
2 1, a wooden stick is 8 meters long and divided into two sections, one of which is longer than the other 1 meter. In order to find the length of these two sections, suppose that one section is x meters and the other section is y meters, then the binary linear equation of the column is
22. If the circumference of a rectangle is 20 cm and the length ratio is 2 cm, then the length and width of the rectangle are both cm.
23.7 (2) Class has 6 teachers and 30 students, of whom 60% are boys. If you draw a fan-shaped statistical chart of the number of all teachers and students in this class, the central angle of the fan-shaped boys is.
24. Xiaoli used 250 yuan money to buy an item in a supermarket and found that the price of each item was 2.5 yuan/piece, but the promotional advertisement in the supermarket indicated that if she bought more than 100 pieces (excluding 100 pieces), the price was 2.4 yuan/piece, so she could only buy one item at most with her money.
25. A classmate bought 80 cents stamps and 1 yuan stamps for 16 yuan. It is understood that the one-dollar stamp he bought is two less than the eighty-point stamp. If he buys 80 cents stamps, the equation is () according to the meaning of the question.
26. The purchase price of a commodity is 15 yuan, and the price is 22.5 yuan when it is sold. Because the market is depressed and the sales situation is not good, the store is prepared to reduce the price, but to ensure the profit rate is not lower than 10%, then the store will reduce the price by _ _ _ _ _ yuan at most to sell the goods.
27. There was a shop that priced a commodity at the purchase price plus 20%, but it could not be sold. Later, the boss reduced the price in 96 yuan by 20% and sold it quickly. Then the profit and loss of this business is ()
A, earn 6 yuan B, lose money or earn C, lose 4 yuan D, and lose 24 yuan.
28. A prize-winning knowledge contest was organized in the class. Xiaoming bought 30 notebooks and pens with a class fee of 100 yuan. In view of 2 yuan for each notebook and 5 yuan for each pen, Xiao Ming can buy at most one pen ().
a，20 B， 14 C， 13 D， 10。
29. A store can make a profit of 25% by selling a batch of clothes 150 yuan. Find the cost price of this kind of clothes. Let the cost price of this kind of clothing be X yuan, then the equation is ().
a、 150-xx =25% B、 150-x=25% C、x= 150×25% D、25%？ x = 150
30. The school canteen sells two kinds of bread with the same thickness and different sizes. The small cake is 30 cm in diameter and costs 30 cents, while the big cake is 40 cm in diameter and costs 40 cents. You prefer to buy _ _ _ _ _ _ _ _ _ cakes at the price of _ _ _ _ _.
3 1, a book mall offers preferential activities to students, and all books purchased at one time do not exceed 200 yuan will be given a 10% discount, while those in 200 yuan that exceed 200 yuan will be given a 9% discount, and those that exceed will be given a 8% discount. When a student buys a book, he pays 72 yuan once and gets a 20% discount for the second time. He checked the pricing of the books he bought and found that he saved 34 yuan money twice. Then the student actually paid _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ yuan for the second book purchase.
32. A raw material supplier implements the following preferential measures for customers who purchase its raw materials: (1) No discount will be given if the one-time purchase amount does not exceed 1 10,000 yuan; (2) A one-time purchase of more than 6,543,800 yuan, but not more than 30,000 yuan, enjoys a 10% discount; (3) If the one-time purchase amount exceeds 30,000 yuan, 30,000 yuan will be given a 10% discount, and the part exceeding 30,000 yuan will be given a 20% discount. A factory paid 7800 yuan for the first purchase of raw materials from suppliers and 26 100 yuan for the second purchase due to inventory reasons. If he buys the same raw materials at one time, he can pay less ()
A, 1460 yuan b, 1540 yuan c, 1560 yuan d, 2000 yuan.
33. In the seventh grade football round robin, it is stipulated that the winner will get 3 points, the draw will get 1 point, and the loser will get 0 point. Now Class 7 (I) played 8 games and got 19 points. So the status quo of Class Seven (1) is _ _ _ _ _ _ _ _ _.
Knowledge point 2 ancient problems
1. Old question: "I asked Li Sangong about opening a store, and all the guests came to the store. There are seven guests in one room, nine guests in one room, and one room is empty. " Then there were _ _ _ _ _ rooms with _ _ _ guests.
2. Today, there are large and small rice barrels, 5 large barrels plus 1 small barrel, which can hold 3 hectares of rice; 1 The vat and five small vats can hold 2 tons of rice. Find out how many meters the big barrel and the small barrel are filled with (call: the name of the ancient measuring instrument). If there are x tons of rice in a big barrel and y tons of rice in a small barrel, the equation can be listed as _ _ _ _ _ _ _.
3. "Today, the chicken and the rabbit are in the same cage, with 35 heads above and 94 feet below. Ask chickens and rabbits about their geometry. " Topic: At present, chickens and rabbits are in the same cage, with 35 heads above and 94 feet below. How many chickens and rabbits are there?
There are some mathematical problems written in fairy tales in Greek anthology. For example, the topic of mule carrying goods was once adapted by the great mathematician Euler. The topic is this: the donkey and mule are carrying goods side by side on the road, and the donkey keeps complaining that the goods they carry are too heavy to bear. The mule said to the donkey, "What are you complaining about?" The goods I brought are heavier than yours. If you give me a bag of goods, I will carry twice as much goods as you. If I give you a bag, we will carry the same amount of goods. "How many bags of goods can donkeys and mules carry?
Can you solve this problem with equations?
◆ General application problems of regularization method
There are three fewer basketball teams in our school than there are twice as many football teams, and the ratio of basketball to football is 3: 2. How many people are there in these two teams?
There are 48 teams and 520 athletes participating in a basketball and volleyball match, including basketball team 10 and volleyball team 12. How many teams are taking part in basketball and volleyball matches?
There are 48 teams and 520 athletes participating in a basketball and volleyball match, including basketball team 10 and volleyball team 12. How many teams are there in basketball and volleyball?
There are two kinds of metals, A and B. 16 parts of A is equal to 33 parts of B, and 55 parts of B is 7 grams heavier than 40 parts of A.. How many grams of each metal do you want?
The second workshop of a factory has 30 fewer people than the first workshop. If 10 people are transferred from the first workshop to the second workshop, the number of people in the second workshop is three quarters of that in the first workshop. How many people are there in each of these two workshops?
Xiao Li is one fifth of his grandfather's age this year. Xiao Li found that after 12, his age became one third of his grandfather's. Try to find out Xiao Li's age this year.
Xiaoming and Xiao Liang play addition games. Xiao Ming wrote an extra 0 after an addend, and the total is 242. Xiao Liang wrote an extra 0 after one addend, and the total was 34 1. What are the first two addends?
(Sum difference, trip problem) A highway of more than 5 meters was built in the first day of junior high school; The next day, the whole road was repaired by a fifth, leaving 14 meters. How long is this road?
An old man cut a long straw rope into front, middle and back sections. The middle section is equal to the front section length plus the rear section length, and the rear section length is equal to half of the front section length plus the middle section length. Now we only know that the length of the front section is 5m. How long are the middle and back sections of the straw rope?
* * * The Central Committee of the Communist Youth League launched the action of "Protecting Mother River", and two classes of Grade 9 in a school 1 15 students actively participated. It is known that one third of the students in class 9/kloc-0 donated money 10 yuan, two fifths of the students in class 92 donated money 10 yuan, and the students in the other two classes donated money 10 yuan.
(Sum and difference problem) A testing station should test a batch of instruments within a specified time. It was originally planned to test 30 such instruments every day, but only three-sevenths of the total can be detected within the specified time; Now 40 sets are actually tested every day. As a result, the task was not only completed one day ahead of schedule, but also 25 more tests were conducted. How many days is the specified time? How many sets of instruments are there in this batch?
There are a group of children in the swimming pool. Boys wear blue swimming caps and girls wear red swimming caps. If every boy sees as many blue swimming caps as the red ones, and every girl sees 1 times as many blue swimming caps as the red ones, do you know how many boys and girls there are?
Question: (1) What is the known quantity in the question? What is the required quantity?
(2) What are the equal relations?
(Travel problem) A ship sails downstream at a speed of 20 kilometers per hour; Countercurrent sailing speed 16 km. So this ship is traveling at a speed of several kilometers per hour in still water?
(Travel problem) A does aerobic exercise at a speed of 5km/h, and after 2 hours, B rides a bicycle to catch up with A from the same place along the same road. According to their agreement, if B catches up with A no earlier than 1h and catches up with A no later than 1h 15min, what range should B control the riding speed?
(Travel problem) The road from A to B has an uphill section, a flat road and a 3km-long downhill section. If you go uphill at 3 kilometers per hour,
Walking 4 kilometers per hour on a flat road and 5 kilometers per hour on a downhill road, then it takes 90 minutes to get from A to B, 102 minutes to get from B to A, and the whole journey from A to B.
how much is it?
(Travel problem) A class of students went to Beishan for an outing, 18km. There is only one car, so it needs to be divided into two groups. Group A takes the bus first, and Group B walks. When the bus arrives at A, Group A gets off and walks, and the bus returns to Group B. Finally, the two groups arrive at Beishan Station at the same time. Given the speed of 60 km/h and the walking speed of 4 km/h, find the distance from point A to Beishan.
(Travel problem) Both parties set out from two places at the same time, and the first meeting was 50 meters away from the midpoint. After they arrived at two places, Party A and Party B immediately turned around and walked back. So, Party A and Party B met for the second time at a distance of 0/00 meters from Party A/KLOC-to find the distance between Party A and Party B. ..
(Travel problem) Both parties set out from two places at the same time, and the first meeting was 50 meters away from the midpoint. When Party A and Party B arrive at the two places, they immediately turn back and walk back. So, Party A and Party B met for the second time at a distance of 0/00 meters from Party A/KLOC-to find the distance between them.
(Travel problem) Two trains leave from two opposite places 9 10 kilometers apart at the same time, and meet in 10 hours. If the first train leaves 4 hours and 20 minutes earlier than the second train 1, then meet after the second train leaves 8 hours and find the speed of the two trains.
(Travel problem) A class of students went to Beishan for an outing, 18km. There is only one car, so they have to be divided into two groups. Group a takes the bus first, and group b walks. When the bus arrives at A, Group A gets off and walks, and the bus returns to meet Group B. Finally, the two groups arrive at Beishan Station at the same time. Given the speed of 60km/h and the walking speed of 4km/h, find the distance from point A to Beishan Station.
(Travel problem) Correspondents should arrive at a certain place within the specified time. If they take15km/h, they can arrive at a certain place 24 minutes in advance. If you walk 12 kilometers per hour, you will be late 15 minutes. How many kilometers does it take for a reporter to reach a certain place? And how many hours was the initial time?
First-year students go to the canteen for a meeting. If every four people sit on a bench, 28 people have no place to sit. If six people are sitting on a bench, find out the number of first-year students and benches.
(Distribution and transportation) Two batches of goods were transported to the disaster area. The first batch of ***480 tons has just been transported in 8 train cars and 20 cars; The second batch of * * * shipped 524 tons, which has just been shipped 10 train cars with 6 cars. How many tons is each train car and each car loaded on average?
Some students live in dormitories. If there are 4 people in each room, that's 20 people. If there are 8 people in each room, one room will be empty and not satisfied. How many dormitories and students are there?
Distribute several exercise books to several students. If each student is divided into four books, there are still 20 books left. If each student gets 8 copies, then the last student gets less than 8 copies. Ask the number of students and the number of exercise books.
(Distribution problem) In the extracurricular reading class, the teacher distributed 43 books to each group, with 8 books in each group and the rest. Nine books per group is not enough. How many groups are there?
Xiaolong and Xiaogang are playing pinball. Xiaolong said to Xiaogang, "Give me half of your beads, and I will have 10 beads." Xiao Gang said, "If you give me yours, I will 10." If Xiaogang's number of marbles is one, Xiaolong's number of marbles is one.
Xiaoming plays basketball with his father. The rules they agreed were: Xiao Ming scored 1 to get 3 points, and Xiao Ming's father scored 1 to get 1. As a result, they hit 20 * *. After calculation, it was found that their scores were completely equal. Can you tell me how much they played?
(Distribution problem) The first two batches of goods, 480 tons, shipped to the disaster area have just been transported in 8 train cars and 20 cars; The second batch of * * * shipped 524 tons, which has just been shipped 10 train cars with 6 cars. How many tons is each train car and each car loaded on average?
First-year students go to the canteen for a meeting. If every four people sit on a bench, 28 people have no place to sit. If six people are sitting on a bench, find out the number of first-year students and benches.
(Distribution problem) Making tin cans. Each piece of iron sheet can be made into 16 boxes, or 43 boxes at the bottom, and one box body and two boxes at the bottom can be made into a set. At present, there are 150 tinplates. How many sheets are used to make the box and how many sheets are used to make the bottom of the box, which can just match?
(Distribution problem) A workshop originally planned to produce 165 parts in 30 days. In the first eight days, * * * produced 52 parts. Due to the adjustment of construction period, it is required to overfulfil the task five days in advance. How many parts will be produced on average every day in the future?
(Distribution) The main players of a basketball team made 28 of 22 shots in a 14 match. Besides three three-pointers, he also hit two points.
How many balls and free throws are there?
(Distribution problem) A group of girls live in several dormitories, each with 4 people, and the remaining 9 people have no room to live; There are 6 people in each room, and one dormitory is not enough. How many dormitories and students are there?
There are now 400 machine parts to be processed. If A does 1 day first, and then two people do it for 2 days, there are still 60 unfinished parts. If two people cooperate in Qixin for 3 days, they can produce 20 more. How many parts do Party A and Party B produce every day?
Analysis: working hours × working efficiency = workload
(distribution and transportation) A motorcade carries a batch of goods. If each ship is loaded with 50 tons, there are still 25 tons left. If each ship is loaded with 5 tons more, there are still 35 tons of vacancies. How many ships and tons of cargo are there in this fleet?
A transportation company has two kinds of trucks, two big cars and three small cars can transport 15.5 tons, five big cars and six small cars can transport 35 tons, and customer Wang has 52 tons of goods. It is required to transport the goods in the same number of trucks at one time. How many trucks do you need?
(Distribution engineering problem) Both parties are processing a batch of parts at the same time. In the first three hours, they * * processed 126 pieces, and in the last five hours, Party A spent 1 hour repairing tools, so Party A processed 10 pieces per hour more than before. Results Party A processed 10 pieces more than Party B in the later period.
(Distribution geometry) Using rectangular and square cardboard as shown in figure 1 as the side and bottom, two kinds of open cartons are manufactured as shown in figure 2, vertical and horizontal. At present, there are 1500 square cardboard and 100 1 rectangular cardboard in the warehouse. How many cartons are made for each of the two models, just to make the cardboard in stock run out?
(Financial problems) There are three kinds of packaging for a beverage, and 1 medium bottle is 20 cents cheaper than 2 small bottles, and 1 large bottle is 40 cents more expensive than 1 medium bottle and 1 small bottle. How much does it cost to buy 1 bottle in large, medium and small bottles, 9 yuan and 60 cents?
(financial problems) 5. In the first phase, a shopping mall engaged in preferential sales promotion, and it was decided that customers would draw lots to decide the discount. A customer bought two kinds of goods, A and B, which were discounted by 30% and 10% respectively, and * * * paid 368 yuan. The sum of the original prices of these two commodities is 500 yuan. What are the original prices of these two commodities?
(Financial problems) A factory bought 56 tons of materials A and B, which cost 9860 yuan. If material A is per ton 190 yuan and material B is per ton 160 yuan, how many tons will they buy?