Several solutions of linear equation with one variable;
1. Denominator: After observing the composition of the equation, multiply the left and right sides of the equation by the least common multiple of each denominator.
2. Remove the brackets: After carefully observing the equation, remove the brackets in the equation, then remove the brackets, and finally remove the braces.
3. Move the term: move all the unknown terms in the equation to the other side of the equation, and move all the remaining terms to the other side of the equation.
4. Merging similar terms: By merging the same terms in the equation, the equation is transformed into the form of ax=b(a≠0).
5. Change the coefficient into 1: divide the two sides of the equation by the unknown coefficient A, and the coefficient before x becomes 1, and get the solution of the equation.
Application of one-dimensional linear equation:
One-dimensional linear equation can solve most engineering problems, travel problems, distribution problems, profit and loss problem, integral table problems, telephone billing problems and digital problems. If only arithmetic is used, some problems may be extremely complicated and difficult to understand.
The establishment of linear equation model will be able to find the equivalence relationship from practical problems and abstract it into a mathematical problem that can be solved by linear equations.
For example, it is proved that "the cycle of 0.9 is equal to 1" within the scope of elementary mathematics. By verifying the rationality of the solution of linear equation, the purpose of explaining and solving life problems is achieved, and some problems in production and life are solved to some extent.
What is a linear equation with one variable? An equation contains only 1 unknowns, the highest number of unknowns is 1, and both sides are algebraic expressions [it must be satisfied that it contains unknowns and both sides are algebraic expressions], which is called a linear equation with one variable.
The expression of linear equation of one variable: ax+b=0, where a≠0.
For example, 3x+5= 1 1 is a linear equation.
3x+5 is not a linear equation because it is not an equation.
3×2+5= 1 1 is not a linear equation because there are no unknowns.
1+5= 1 1 in x is not a linear equation, because neither side of the equation is an algebraic expression.
3x_+5= 1 1 is not a linear equation because the degree of the highest term is not 1.
The general steps to solve a linear equation with one variable are:
Denominator: Multiply both sides of the equation by the least common multiple of each denominator.
Remove brackets: first remove brackets, then remove brackets, and finally remove braces.
Move term: move all the terms containing unknowns to one side of the equation and all other terms to the other side of the equation.
Merge similar terms: transform the equation into ax[+c]=b(a≠0).
Coefficient becomes 1: divide the unknown coefficient a on both sides of the equation to get the solution of equation x = b [-c]/a.
take for example
3x+5= 1 1
Solution [required]: 3x+5-5 = 1 1-5.
3x=6
3x÷3=6÷3
x=2
Eight common formulas for solving the application problems of linear equations with one variable
① Sum, difference, multiplication and division, that is, the sum of two numbers = larger number+smaller number, and larger number = smaller number × multiple addition and subtraction;
② Travel problem, that is, distance = speed × time;
(3) engineering problems, that is, workload = work efficiency × working hours;
④ Concentration problem, that is, solute mass = solution mass × concentration;
⑤ Distribution problem, that is, the total amount remains unchanged before and after deployment, and a new proportional relationship appears between the two sides after deployment;
⑥ Equal product problem, that is, the mass (or volume) is unchanged before and after deformation;
⑦ Number problem, that is, if the number on one bit is A, the number on the tenth bit is B and the number on the hundredth bit is C, these three bits can be expressed as100c+10b+a;
⑧ Economic problems, that is, interest = principal × interest rate × number of periods; Sum of principal and interest = principal+interest = principal+principal × interest rate× number of periods; After-tax interest = principal × interest rate × number of periods ×( 1- interest tax rate); Commodity profit = commodity price-commodity purchase price; Commodity profit rate = profit × 100%.
Hope to adopt
What is a linear equation with one variable? There is only one unknown, the higher order of the unknown is 1, and both sides of the equal sign are algebraic expressions. Such an equation is called a linear equation. Its general form is:
One-dimensional linear equations were first seen in ancient Egypt around 1600 BC. Around 820 AD, the mathematician Hua La Zi Mi put forward the idea of "merging items" and "shifting items" in his book "Eliminating Yuan and Reducing Yuan". In16th century, after the mathematician David founded symbolic algebra, he put forward the propositions of shifting terms and division of the same equation. 1859, mathematician Li officially translated this kind of equations into linear equations.
Extended data:
There are five steps to solve a linear equation with one variable, namely, denominating, removing brackets, shifting terms, merging similar terms, and the transformation coefficient is 1. All steps are carried out according to the properties of algebraic expressions and equations.
To solve this equation
For example:
1, denominator, get:
2. Remove the bracket to obtain:
3. Move and get:
4. By merging similar projects, we can get:
5, the coefficient becomes 1, and we get:
Baidu encyclopedia-unary linear equation
What is a linear equation with one variable? In an equation, if there is only one unknown linear equation, and the highest order of the unknown linear equation is 1, then the whole equation is called a linear equation. (One-dimensional linear equation)
General form: ax+b=0(a and b are constants, a≠0). One-dimensional linear equation has only one solution.
The final result of a linear equation (the solution of the equation) is in the form of x = a.
"Properties of Equation 1" and "Properties of Equation 2" for linear equations with one variable.
1. Both sides of the equation add and subtract the same number at the same time, and both sides of the equation are equal. (if a=b, then a c = b c )
2. Both sides of the equation are multiplied or divided by the same number (except 0) or an algebraic expression, and the two sides of the equation are equal. (if a=b, then ac=bc. If a=b and c≠0, then a/c = b/c. )
The solution is to move the unknown to one side by shifting the term, and then move the constant to one side (the basic property of the equation is 1, pay attention to the symbol! ), and then divide the two sides by the unknown coefficient (the coefficient is 1, and the basic property of the equation is 2), and the unknown value can be obtained.
For example: 7x+23= 100
Solution: 7x= 100-23
7x=77
x = 77 \7
x= 1 1
In elementary school arithmetic, we learned the knowledge of solving practical problems with arithmetic. So, can a linear equation solve a practical problem? If it can be solved, how? What are the advantages of solving application problems with one-dimensional linear equations compared with solving application problems with arithmetic methods?
To answer these questions, let's look at the following examples.
Example 1 3 times of a certain number minus 2 equals the sum of a certain number and 4, so find a certain number.
(First, solve it by arithmetic, the students answer, and the teacher writes it on the blackboard.)
Solution 1: (4+2) ÷ (3- 1) = 3.
A: A certain number is 3.
(Secondly, solve the problem by algebraic method, with the guidance of the teacher and oral completion by the students. )
Solution 2: Let a certain number be x, then there is 3x-2 = x+4.
X = 3 is obtained by solving.
A: A certain number is 3.
Looking at the two solutions of the example 1, it is obvious that the arithmetic method is not easy to think about, but the method of setting unknowns, listing equations and solving equations to solve application problems has a feeling of making it difficult, which is also one of the purposes of learning to solve application problems with linear equations.
We know that the equation is an equation with unknowns, and the equation represents an equal relationship. Therefore, for any condition provided in an application problem, we must first find an equal relationship from it, and then express this equal relationship as an equation.
Simple application: find addend = sum-another addend.
Find the minuend = difference+subtraction
Subtraction = minuend-difference
Factor = product/another factor
Find dividend = quotient * divisor
Find divider = divider/quotient
Universal solution:
1. Both sides of the denominator equation are multiplied by the least common multiple of each denominator.
4. Generally, remove the brackets first, then remove the brackets, and finally remove the braces. But sometimes the order can be determined according to the situation, which makes the calculation simple. According to the law of multiplicative distribution.
3. Move the unknown term to the other side of the equation, and don't forget to change the sign when moving other terms to the other side of the equation.
4. Merge similar terms to transform the original equation into ax=b(a≠0).
5] coefficient 1 equation, both sides are divided by unknown coefficient at the same time, and the solution of the equation is obtained.
One-dimensional linear equation exercise
Basic question:
First, multiple-choice questions:
1, which belongs to the linear equation in the following categories is ().
A.5a+4b B.4x+9x
C.5x2+9y2 D. 7a-4b
2. The solution of equation 3x-2=-5(x-2) is ().
A.- 1.5 b . 1.5 c . 1d .- 1
3. If the solution of the equation about satisfies the equation, the value of is ().
A. 10 8th century BC.
4, according to the following equation property is correct ()
A. Fuck, get B. Fuck, get
C. pass, get D. pass, get
5. When solving the equation, after removing the denominator, the correct result is ()
A.B.
C.C.
6. The price of TV sets has been reduced twice in a row 10%. After the price reduction, the price of each TV set is one yuan, so the original price of TV set is ().
A.0.8 1a yuan B. 1.2 1a yuan C. 1. 1a yuan D.0. 1a yuan.
8. A store sells two pieces of clothes, each 60 yuan, one earns 25% and the other loses 25%. Then after these two clothes are sold, the store is ().
A. Don't earn or lose B. Earn 8 yuan C. Lose 8 yuan D. Earn 8 yuan
9, in the following equation, belongs to the linear equation is ().
(A) (B) (C) (D)
10, the solution of the equation is ()
(A) (B) (C) (D)
1 1, given the equation, the following equation may not be true ().
(A) (B)
(C) (D)
12, the solution of the equation is that it is equal to ()
(A) (B) (C) (D)
13, solve the equation, remove the denominator, and get ()
(A) (B)
(C) (D)
14, the following equation variation, the correct is ().
(1) Equation, shift term, get
(b) Equation, without brackets, gives
(c) Equation, which converts the unknown coefficient into 1 to obtain
Formation of equation
15. Son 12 years old, father 39 years old. The father is four times as old as his son.
(a) Three years later; (B)3 years ago; Nine years later; No way.
16. The football used by chongqing lifan New Feeling Football Team for training is made of 32 pieces of black and white cowhide, of which the black leather can be regarded as a regular pentagon and the white leather as a regular hexagon. The ratio of black and white leather blocks is 3:5, and the number of black and white leather blocks is required. If the number of black leather blocks is 0, the listed equation is correct ().
(A) (B)
(C) (D)
17, Coral Middle School built a comprehensive building, leaving a rectangular open space with a length of more than 5m and a circumference of 50m. In order to beautify the environment, the school decided to plant it as turf. It is known that the minimum planting cost per square meter of turf is RMB, so planting turf needs at least ().
(1) Yuan; (b) yuan; (c) yuan; (4) yuan.
One year, two years, three years.
2.25 2.43 2.70
18. The annual interest rate of bank education savings is shown in the following table:
Xiaoming is in grade seven now. In July this year, his parents deposited 30 thousand yuan in the bank for him to use when he was in high school three years later. After three years, the income will be maximized, and Xiao Ming's parents should adopt ().
(a) Direct deposit for three years;
(b) Deposit first 1 year, and after 1 year, the total interest will be automatically transferred to the 2-year period;
(c) Deposit one 1 year, and after 1 year, automatically transfer the principal and interest to two 1 year;
(d) Deposit a 2-year deposit first, and after 2 years, the total interest will be automatically transferred to 1 year deposit.
2. Fill in the blanks:
1, then _ _ _ _ _.
2, known, then _ _ _ _ _ _ _.
3. If the solution of the equation is 3, the value of is _ _ _ _ _ _ _ _.
4. If there is a three-digit number, its unit number is, the number on the tenth digit is, and the number on the hundredth digit is, then this three-digit number is expressed as _ _ _ _ _ _ _ _ _ _ _.
5. There are 96 people in Class A and Class B, and Class A has 2 more people than Class B, so Class B has _ _ _ _ _ _ _ _.
6. A number is twice as big as its half. If a number is set to, the equation is _ _ _ _.
7. When _ _, the values of algebraic sum are opposite.
8, formula, known, then _ _.
Day one two three four five six
1 2 3 4 5 6
7 8 9 10 1 1 12 13
14 15 16 17 18 19 20
2 1 22 23 24 25 26 27
28 29 30 3 1
9. As shown on the right, it is the calendar of 65438+February 2003. At present, four numbers are arbitrarily framed by a rectangle in the calendar.
Please use an equation to express the relationship between _ _ _ _ _ _ _ _.
10, a long cylindrical test tube with an inner diameter of 3_ filled with water. Now gradually drop the water in the test tube into a cylindrical glass with an inner diameter of 8_ and a height of 1.8_. When the glass is filled with water, the water in the test tube drops _ _ _ _.
1 1. During the National Day, New Century Department Store has a seasonal discount. Jane bought a sportswear at a 20% discount, saving 16 yuan, so he actually spent _ _ _ yuan on this dress.
12, the total length of chengdu-chongqing railway is 504 kilometers. An express train leaves Chongqing at a speed of 90 kilometers per hour. 1 hour later, another local train departed from Chengdu at a speed of 48 kilometers per hour. Then the two trains meet _ _ hours after the local train leaves (regardless of the stop time along the way).
13. When we were young, we heard the story of the race between the tortoise and the hare. We all know that the tortoise finally defeated the white rabbit. If in the second race, the white rabbit is ashamed and brave, and catches up with the tortoise at the speed of 10 1 m/min when it is behind the tortoise, and the tortoise still crawls at the speed of 1 m/min.
14, the annual interest rate of one-year fixed deposit is 1.98%, and 20% interest must be deducted as interest tax and turned over to the state treasury. If Xiaoying deposits a one-year time deposit, and the interest after tax is 158.4 yuan, then the RMB she deposits is _ _ _ _ _
15, 52 cars lined up in two teams, each car was one meter long, and the distance between the two cars was 3a/2m. The average speed of the team was 50 meters per minute. It takes 16 minutes for this train to pass through the 546m long square, so a = _ _ _ _ _ _ _
Third, solve the equation:
1、 2、
3、 4、
5、 6、
7、 8、
9. Knowing the root of the equation, find the value of the algebraic expression.
Four, column equation to solve application problems:
1. The enemy escaped at the station 8 kilometers away from our army. The time is 4 am. Our army set off at 5 o'clock and pursued at the speed of 10 km per hour. And I caught up at 7: 00. What is the speed at which the enemy fled?
In the mid-term exam, the information technology teacher asked every seventh-grade student to write an article within 40 minutes. As we all know, it takes Bao Xiao 50 minutes to finish an article of the same size independently, while Beckham only needs 30 minutes. In order to complete the task, Bao Xiao asked Beckham for help and cooperation after playing for 30 minutes. Can he finish it in the required time?
3. After learning "Operation of Rational Numbers", five students from each class of the seventh grade in the experimental middle school were selected to form a team, and a knowledge contest was held under the organization of the math teacher. The rules of the game are: each team will give 50 questions respectively, and get 3 points for answering one question correctly. (1) If the team in Class Two finally got 142, how many questions did the team in Class Two answer correctly? (2) Can the final score of the first team be 145? Please briefly explain the reasons.
A "Hope School" has built a four-story teaching building with six classrooms on each floor, and there are three doors (two front doors and a side door of the same size) in and out of the building. During the safety inspection, these three doors were tested: when a front door and a side door are opened at the same time, 400 students can pass through in 2 minutes, and if a front door can pass through 40 more students per minute on average than a side door.
(1) How many students can pass through a main entrance and a side door every minute on average?
(2) During the inspection, it was found that the efficiency of going out was reduced by 20% due to the crowded students in an emergency. According to the safety inspection regulations, in case of emergency, students in the whole building should be evacuated safely through these three doors within 5 minutes. Suppose there are at most 45 students in each classroom of this teaching building. Q: Do the three doors meet the safety regulations? Why?
5. Mother Black Bear wanted to test the effect of Little Bear's learning "solving application problems with equations", and gave him 19 apples to divide him into four piles. After dividing, if the first pile is doubled, the second pile is increased by one, the third pile is reduced by two and the fourth pile is doubled, the number of apples in these four piles will be the same. How does the bear divide these 65400 apples?
6. The school is going to spend 2000 yuan to buy prizes for 22 students who won the Hope Cup competition. The first prize was awarded to 200 yuan, and the second prize was awarded to 50 yuan. How many students will win the first prize and the second prize respectively?
7. A store will increase the cost price of a commodity by 40%, and then mark it. During the new year's day, it wants to sell at a 20% discount to thank new and old customers for their patronage. The price is 224 yuan. What is the cost price of this commodity?
8. Party A and Party B visited the exhibition hall which is 0/000 meters away from the school. Party A leaves for 5 minutes before Party B leaves. The speed of Party A is 80m/min, and the speed of Party B is 180m/min. How long will it take Party B to catch up with Party A? How far is it from the exhibition hall when we catch up with A?
Higher requirements:
1, known, then the algebraic value.
2.(200 1 Senior High School Entrance Examination in Wuxi City, Jiangsu Province) According to the market information, a shopping mall has adjusted the prices of two existing air conditioners of different models. One air conditioner sold after price adjustment can make a profit of 10% (relative to the purchase price), while another air conditioner sold after price adjustment will lose 10% (relative to the purchase price).
(a) neither making a profit nor losing money; (b) 1% profit; 2% loss; 1% loss.
The developer sells the house by installment. Xiao Ming's family bought a new house, and the current price is 1.2 million. When buying a house, they need to pay a down payment of 30,000 yuan (the first year). Starting from the second year, the house payment should be 5000 yuan per year plus the interest on the remaining arrears of the previous year. It is known that the annual interest rate of the remaining amount is 0.4%. In what year does Xiaoming's family need to pay 5200 yuan?
A milk processing factory has 9 tons of fresh milk. If you sell fresh milk directly in the market, you will make a profit of 500 yuan per ton. If you sell yogurt, the profit per ton 1200 yuan. If it is made into milk tablets for sale, it can make a profit of 2000 yuan per ton.
Option 1: make as many pieces of milk as possible and sell the rest directly;
Option 2: part of it is made into milk tablets, and the rest is made into yogurt for sale, which is completed in exactly 4 days;
(1) Which scheme do you think is the most profitable? Why?
(2) What questions can be asked after this problem is solved? If not, write down your question?
Two cars start from the same place at the same time and drive straight in the same direction at the same speed. Each car can only take 24 barrels of gasoline at most, and no other oil can be used on the way. A barrel of oil can propel a car 60 kilometers forward. Both cars must return to the starting point, but they can return at different times. Both cars can borrow oil from each other. In order to make one car as far away from the starting point as possible, how many kilometers should the other car return from the starting point? How many kilometers has the car farthest from the starting point traveled?
What is a linear equation with one variable? One-dimensional linear equation refers to an equation with only one unknown number, the highest order of which is 1, and both sides are algebraic expressions. A linear equation with one variable has only one root. One-dimensional linear equation can solve most engineering problems, travel problems, distribution problems, profit and loss problem, integral table problems, telephone billing problems and digital problems.
Solutions of one-dimensional linear equations
1, denominator: both sides of the equation are multiplied by the least common multiple of each denominator;
2. Remove the brackets: first remove the brackets, then remove the brackets, and finally remove the braces;
3. Move the term: move all the terms containing unknowns to one side of the equation, and all the other terms to the other side of the equation; Move the item to change the symbol.
4. Merge similar terms: transform the equation into ax = b (a ≠ 0);
5. The coefficient becomes 1.
The history of one-dimensional linear equations
One-dimensional linear equations were first seen in ancient Egypt around 1600 BC. Around 820 AD, the mathematician Hua La Zi Mi put forward the idea of "merging items" and "shifting items" in his book "Eliminating Yuan and Reducing Yuan". In16th century, after the mathematician David founded symbolic algebra, he put forward the propositions of shifting terms and division of the same equation. 1859, mathematician Li officially translated this kind of equations into linear equations.