Chapter 7 Binary linear equations and their applications (1)
First, multiple choice questions
1.(20 12? Texas) is known, then a+b is equal to ()
A.3B。 2D。 1
Test site: solving binary linear equations.
Special topic: calculation problems.
Analysis: ①+② We get 4a+4b= 12, and divide both sides of the equation by 4 to get the answer.
Answer: Solution:,
∫①+②:4a+4b = 12，
∴a+b=3.
So choose a.
Comments: The key to understanding the application of binary linear equation is to examine whether students can use ingenious methods to get answers. The topic is typical and good.
2.(20 12 Heze) is known as the solution of binary linear equations, then the arithmetic square root of ().
A.2b . 2 c . 2d . 4
Test site: the solution of binary linear equations; Arithmetic square root.
Solution: Solution: ∫ is the solution of binary linear equations.
∴ ,
Solution:
∴2m﹣n=4，
The arithmetic square root of ∴ is 2.
So choose C.
3. (Binzhou, 20 12) Li Ming goes to school by bike in the morning, because it takes a long way to build roads. It took him 15 minutes to get to school. The average speed of his cycling is 250m/min, and the average speed of his walking is 80m/min. The distance from his home to the school is 2900 meters. If he rides and walks separately,
A.B.
C.D.
Test center: abstract binary linear equations from practical problems.
Answer: Answer: It takes him x minutes to ride a bike and y minutes to walk.
Therefore, choose: d.
4.(20 12 Linyi) If the solution of the equation of x and y is yes, the value is ().
A.5 B.3 C.2 D. 1
Test site: the solution of binary linear equations.
Solution: The solution of the equations is,
∴ ,
Solve,
Therefore | m ﹣ n | = | 2 ﹣ 3 | = 1.
So choose D.
5、(20 12? Deyang) In order to ensure information security, information needs to be transmitted in an encrypted way, from plaintext to ciphertext (encryption) at the sender and from ciphertext to plaintext (decryption) at the receiver. The known encryption rules are: plaintext A, B, C and D correspond to ciphertext a+2b, 2b+c, 2c+3d and 4D. Such as plaintext 1, 2, 3.
A.7，6 1，4b。 6.4 1.7c。 6， 1，7d 1，6，4，7
Test site: the application of binary linear equations.
Analysis: Know the result (ciphertext), find the plaintext, and solve the column equations according to the rules.
Answer: Answer: According to the meaning of the question, you must
,
Solve.
∴ Plain text: 6,4, 1, 7.
So choose B.
Comments: This question examines the application of the equation in practice and clarifies the meaning of the question. Establishing equations is the key to solving problems.
6.(20 12? Hangzhou) on the known equations of x and y, where -3 ≤ A ≤ 1, the following conclusions are given:
① is the solution of equations;
② When a =-2, the values of X and Y are reciprocal;
③ When a= 1, the solution of the equations is also the solution of the equation X+Y = 4-A;
④ If x≤ 1, then 1 ≤ Y ≤ 4.
The correct one is ()
A.①② B.②③ C.②③④ D.①③④
Test site: the solution of binary linear equations; Solve a system of linear inequalities with one variable.
Analysis: Solve the equations to get the expressions of X and Y, determine the range of X and Y according to the range of A, and judge them one by one.
Solution: solution: solve the equation and get,
∵﹣3≤a≤ 1,∴﹣5≤x≤3,0≤y≤4，
① Inconsistency-5 ≤ x ≤ 3, 0≤y≤4, and the conclusion is wrong;
② When a=﹣2, x= 1+2a=﹣3, y= 1﹣a=3, and the values of x and y are reciprocal, the conclusion is correct;
③ When a= 1, X+Y = 2+A = 3, 4-A = 3, and both sides of the equation X+Y = 4-A are equal, the conclusion is correct;
④ when x≤ 1, 1+2a≤ 1, the solution is a≤0, Y = 1-A ≥ 1, and it is known that 0≤y≤4.
So when x≤ 1, 1≤y≤4, the conclusion is correct.
So choose C.
Comments: This topic examines the solution of binary linear equations and the solution of univariate linear inequality. The key is to find the expressions of x and y and the range of values of x and y according to the conditions.
7. (Liangshan Prefecture, 20 12) Yaxi Expressway was officially opened to traffic on April 29, 20 12, with a total length of 420km from Xichang to Chengdu. A car and a bus leave from Xichang and Chengdu in opposite directions at the same time. After 2.5 hours of meeting, when they met, the car traveled 70km more than the bus, and the average speed of the car and the bus was km/h respectively.
A.B.
C.D.
Answer: d
8. (Wenzhou, 20 12) Ticket prices for a scenic spot in nanxi river: 70 yuan for adults and 35 yuan for children. Xiaoming bought 20 * * * tickets and spent 1225 yuan, assuming there are adult tickets and children tickets. According to the meaning, the following equation is correct ().
A.B.
C.D.
Answer: b
Second, fill in the blanks
1.(20 12 Zhanjiang, Guangdong) Please write a binary linear equation system and make it solve as.
Analysis: The answer to this question is not unique. For example,
,
①+②: 2x=4，
Solution: x=2,
Substitute x=2 into ① to get y =- 1,
The solution of binary linear equations is:
So, the answer is: the answer to this question is not unique, for example.
2.(20 12 Guangdong) If both x and y are real numbers and satisfy | x | 3 |+= 0, then the value of () 20 12 is 1.
Test center: non-negative nature: arithmetic square root; The nature of non-negative number: absolute value.
A: A: According to the meaning of the question,
Solution:.
Then () 2012 = () 2012 =1.
So the answer is: 1.
3. The point (x, y) whose coordinate is (2012 Anshun) is the solution of the system of equations is in the first quadrant.
Test sites: univariate function and bivariate linear equation (group).
Answer: Solution:,
①+②，2y=3，y=，
If y= is substituted into ① and =x+ 1, the solution is: x=,
Because 0, > 0,
According to the coordinate characteristics of points in each quadrant,
So the point (x, y) is in the first quadrant of the plane rectangular coordinate system.
So the answer is: 1.
(20 12 Changsha, Hunan) If real numbers A and B satisfy | 3a- 1 |+B2 = 0, the value of ab is 1.
Solution: according to the meaning of the question, 3a- 1 = 0, b=0,
The solution is a=, b=0,
ab=( )0= 1。
So the answer is: 1.
5.(20 12? The solution of Lianyungang) equations is.
Test site: solving binary linear equations.
Special topic: calculation problems.
Analysis: y can be eliminated by ①+②, so that X can be found. Then substitute the value of x into ①, and y can be easily found.
Answer: Solution:,
①+②，get
3x=9，
The solution is x = 3,
Substitute x = 3 into ①, and you get
3+y=3，
The solution is y = 0,
The solution of the original equation is.
So the answer is.
Comments: The key to solve this problem is to master the idea of addition, subtraction, multiplication and division.
6. (Nantong, Jiangsu, 20 12) A movie ticket for 20 yuan and B movie ticket for kloc-0/5 yuan. If you buy 40 A and B movie tickets and use them to go to 700 yuan, you buy 20 A movie tickets.
Application of binary linear equations in test center.
Special application problems.
Let's buy X tickets for movie A and Y tickets for movie B first, and then buy 40 tickets according to the total. After spending 700 yuan, you can get the equations, and then you can get the answer.
Solution: We buy X tickets for movie A and Y tickets for movie B, which is derived from the meaning in the question.
x+y=40
20x+ 15y=700，
Solution: x=20 y=20, which means I bought 20 movie tickets.
So the answer is: 20.
This topic reviews the application of binary linear equations and belongs to the basic topic. The key to solve this problem is to get equations according to the equivalence relationship between meaning and problem.
Third, answer questions.
1.(20 12? Guangzhou) solve the equation.
Test site: solving binary linear equations.
Special topic: calculation problems.
Analysis: According to the reciprocal of the coefficient of y, it can be solved by adding, subtracting and eliminating.
Answer: Solution:,
①+②，4x=20，
The solution is x=5,
Substitute x=5 into ①, 5-y = 8,
The solution is y =-3,
So the solution of the equation is.
Comments: There are two ways to understand the binary linear equation: addition and subtraction and substitution. Choosing addition and subtraction according to the reciprocal of Y coefficient is the key to solving binary linear equations.
2.(20 12 Guangdong) Solution equation:.
Test site: solving binary linear equations.
Solution: Solution: ①+②, 4x=20,
The solution is x=5,
Substitute x=5 into ①, 5-y = 4,
The solution is y= 1,
Therefore, the solution of the inequality group is:
3.(20 12? Qiandongnan prefecture) solving equations.
Analysis:
③+①，3x+5y= 1 1④，
③×2+②，3x+3y=9⑤，
④-① 2y=2, y= 1,
Substitute y= 1 into ⑤, 3x=6,
x=2，
Substitute x=2 and y= 1 into ① to get z = 6-2 × 2-3 × 1 =- 1.
∴ The solution of the equation is.
4.(20 12 Changde, Hunan) Solve the equation:
Investigation of knowledge points: the solution of binary linear equations.
Ability inspection: ① observation ability, ② calculation ability.
Analysis: Through observation, Y is directly eliminated by adding, subtracting and eliminating elements.
Solution: ①+② Get: 3x = 6 ... 6 ...
∴ x=2
X=2 generations ①
∴ y=3
∴ the solution of the equation is
Comments: the idea of solving equations is elimination method, and the methods of eliminating binary linear equations are "substitution elimination method" and "addition and subtraction method"
Destroy Yuan. "
5. (Loudi, 20 12) The sports cultural goods store bought 20 basketballs and volleyballs, and the purchase price and selling price are shown in the table. After all the sales, the profit will be * * * 260 yuan.
Basketball volleyball
Purchase price (RMB/piece) 80 50
Price (RMB/unit) 95 60
(2) Is the profit from selling six volleyballs equal to the profit from selling several basketballs?
Test site: the application of binary linear equations.
Analysis: (1) We buy X basketballs and Y volleyballs, and solve the equations according to the equivalence relation: ① After all 20 basketballs and volleyballs are sold, 260 yuan can make a profit.
(2) Let the profit from selling six volleyballs be equal to the profit from selling one basketball. According to the meaning of the question, we can get the equivalent relationship: the profit of each volleyball ×6= the profit of each basketball ×a, list the equations and get the answer.
Solution: Solution: (1) Suppose you buy X basketballs and Y volleyballs.
Solution:
A: I bought 12 basketballs and 8 volleyballs;
(2) Assume that the profit from selling six volleyballs is equal to the profit from selling one basketball.
6×(60﹣50)=(95﹣80)a，
Solution: a=4,
A: The profit from selling six volleyballs is equal to the profit from selling four basketballs.
Comments: This question mainly examines the application of binary linear equations and linear equations. The key is to find out the meaning of the problem, find out the equivalent relationship in the problem, and list the equations or equations.
6. (Suzhou, Jiangsu, 20 12) China is a country with a serious shortage of fresh water resources. According to relevant data, China's per capita fresh water resources are only those of the United States, and the sum of the per capita fresh water resources of China and the United States is 13800m3. What is the freshwater resource per capita in China and America (unit: m3)?
Test site: the application of binary linear equations.
Special topic: application problem.
Analysis: Let China's per capita freshwater resources be xm3 and the United States' per capita freshwater resources be ym3. According to the equivalence relation mentioned in the question, the equations can be obtained, and the answer can be obtained by solving them.
Solution: Let China's per capita freshwater resources be xm3 and the United States' per capita freshwater resources be ym3. ..
According to the meaning of the question,
Solution:.
A: The per capita freshwater resources of China and the United States are 2300m3 and 1 1500m3 respectively.
Comments: This question examines the application of binary linear equations. The key to solve this problem is to set an unknown number and get equations according to the equivalence relation described in the problem, which is generally difficult.