Example 1: (Beijing People's Education Edition, 2005) In order to save electricity in summer, two measures are often taken: increasing the set temperature of air conditioner and cleaning equipment. At first, a hotel raised the set temperature of A and B air conditioners by 1℃. Results A air conditioner saves 27 degrees more electricity every day than B air conditioner. Then clean the equipment of air conditioner B, so that the total daily electricity saving of air conditioner B is 1. 1 times that of air conditioner A only after the temperature rises, while the specified electricity consumption of air conditioner A remains unchanged, so that the two air conditioners can save 405 degrees of electricity every day. After the temperature is increased by 1℃, how many kWh can each air conditioner save every day?
Analysis: There are four unknowns in this question: the regulated electric quantity of air A after heating, air B after heating, air A after cleaning equipment and air B after cleaning equipment. The equality relationship is as follows: a-a-b-a-b-a = 27, b-a-b =1.1× b-a-b = a-a-b = 405. Class A air conditioners save X degrees of electricity every day, and Class B air conditioners save Y degrees of electricity every day. According to the relationship between the first two and the third equation, the other two unknowns can be expressed, and then according to the relationship between the first and the fourth equation, two binary linear equations are listed to form equations.
Solution: If the temperature is only increased by 1℃, A-type air conditioner will save X degrees of electricity every day, and B-type air conditioner will save Y degrees of electricity every day.
According to the meaning of the question, you must:
Solution:
Answer: Only increase the temperature 1℃, and the A-type air conditioner saves 207 degrees of electricity every day, and the B-type air conditioner saves 180 degrees of electricity every day.
Second, find the length of the line segment with a binary linear equation.
Example 2: (Fengtai District, Beijing, 2005) Make a rectangular floor with eight identical rectangular tiles, and find the length and width of each tile, as shown in the figure.
Analysis: There are two unknowns in this question, namely the length and width of each floor tile. According to the rectangular length of 60, an equation can be obtained. Because the upper and lower edges of the rectangle are equal, an equation can be obtained, thus forming a set of equations.
Solution: Let the length and width of each floor tile be x and y respectively. According to the meaning of the question:
Solution:
Answer: The length of each floor tile is 45 and the width is 15.
Thirdly, the information problem is solved by using binary linear equations.
Example 3: (Rizhao City, 2005) The municipal government held a hearing on tap water price according to social needs and decided to adjust tap water price from April this year. Some information about the adjusted domestic water price is as follows:
Water consumption (m3) Unit price (RMB /m3)
The second part is within 5m3 (including 5m3)
Part x above 5m3.
It is known that Xiaojingjia and Xiao Lei paid 19 yuan and 3 1 yuan respectively in May, and the water consumption of Xiao Lei's family was 1.5 times that of Xiaojingjia.
Please use the above information to find X in the table.
Analysis: According to the water fee paid by Xiaojingjia and Xiao Lei, their water consumption is above 5 m3, and the water consumption is unknown. Therefore, we first set the water consumption of Xiaojingjia in May as y m3, and that of Xiao Lei in May as 1.5ym3 ... countable equations, which are actually binary linear equations about xy and x, and can be solved and then solved.
4. Solving inequality relations by using binary linear equations.
Example 4: (Huzhou City, 2005) An expressway toll station, m (m > 0) Vehicles queue up to wait for the toll to pass. Assuming that the traffic volume (the number of cars passing through the toll booth every minute) is unchanged, the charging speed of each charging window is also unchanged. If you open a toll window, it takes 20 minutes to charge all the cars in line and the cars coming behind. If you open two charging windows at the same time, it only takes 8 minutes to charge all the cars waiting in line and the cars coming behind. If all the cars waiting in line for charging are required to pass within 3 minutes, and the cars coming to the next station are also required to pass at any time, how many charging windows should be opened at least at the same time?
Analysis: There are three unknowns in this question: the number of cars that can be charged per minute, the traffic volume per minute, and how many toll windows need to be opened, but there are only two equal relationships, that is, "If a toll window is opened, it will take 20 minutes to pass all the charges of the cars that were originally queued and later picked up; If you open two charging windows at the same time, it only takes 8 minutes to charge all the cars that were originally queued and later picked up. " There is also an inequality relationship in the topic, that is, "all cars waiting in line for charging are required to pass within 3 minutes, and cars arriving at the station late should pass at any time", then we can list a combination consisting of two binary linear equations and one unary linear inequality. The values of two unknowns are solved by two equations, and finally substituted into inequalities to find out the range of charging windows.
Solution: Assuming that each charging window can charge X cars per minute, and the traffic volume is Y cars per minute, it takes N charging windows to charge all the cars waiting in line within 3 minutes. According to the meaning of the question:
From ① and ②, we can get:, ④.
Substitute ④ into ③ to get:
∵m & gt； 0, ∴n ≥, n is the smallest positive integer, ∴ n = 5.
A: There must be at least five charging windows.
5. Solve a function problem with a binary linear equation.
Example 5: (Heilongjiang, 2005) An enterprise has two cuboid reservoirs A and B, and the water from the first reservoir is injected into the second reservoir at a speed of 6 cubic meters per hour. The figure shows the function image between water depth y (m) and water injection time x (h) in the first and second reservoirs, and answers the following questions in combination with the image:
(1) The functional relationship between water depth y and water injection time x of two reservoirs A and B is obtained respectively.
(2) How long does it take to inject water? The water depths of the two reservoirs A and B are the same;
(3) How long does it take to inject water? The two reservoirs A and B have the same water storage capacity.
Analysis: (1) We can set Y A = K 1x+B 1. Taking (o, 2) and (3, 0) generations, we can get kl=-23, bl=2, ∴ y A =-23x+2.
(2) The water depths of the two reservoirs A and B are required to be the same. In fact, it is to find the intersection coordinates of two linear functions and combine the two linear functions to form a binary linear equation group. The solution of the equations is the coordinates of the intersection of two linear functions. The equations are: x = 35. So after 35 hours of water injection, the water depth of reservoir A and reservoir B is the same.
(3) We can set the bottom area of Reservoir A as S 1 and that of Reservoir B as S2, and the water storage capacity of Reservoir A and Reservoir B is the same within t hours. According to the meaning of the question, it is concluded that 2Sl=3×6, (4- 1)S2=3×6, so Sl=9.
S2=6, and because s1(-23t+2) = S2 (t+1), the solution is t= 1. Therefore, for the water injection of 1 hour, the water storage capacity of reservoir A and reservoir B is the same.
A classic example of one-dimensional linear inequality;
1. The raw material of a factory is 36kg, and the raw material is 20kg. It is planned to use this raw material to produce products A and B 12. It is known that the raw material A 3kg and raw material B1kg are needed to produce a product. It takes 2 kg of raw material A and 5 kg of raw material B to produce a product B. ..
(1) Suppose X pieces of A product are produced, and write the inequality group that X should satisfy;
(2) Please set up several production schemes that meet the meaning of the question.
( 1){ 3X & lt； = 36
{ X & lt= 20
X
(2)
Type b production 12-X pieces.
Then b {2 (12-x) < = 36.
{ 5( 12-X)& lt； =20
= 8 of solution
So 1: A8B4
2: A 9 B3
3 :A 10 B2
4 :A 1 1 B 1
5 : A 12 B0
Example 4 Solving the Problem
(2) Find the non-negative integer solution of inequality 10(x+4)+x≤84.
Analysis: For the (1) small question, it is necessary to understand that "not less than" means "greater than or equal to", and the symbol means "≥"; (2) A small problem is a non-negative integer, that is, an integer in positive number or zero, so the inequality solution of this problem must be a positive integer or zero. Attention should be paid to the correct application of inequality properties in the process of solving problems.
Solution:
∴ 120-8x≥84-3(4x+ 1)
(2)∵ 10(x+4)+x≤84
∴ 10x+40+x≤84
∴ 1 1x≤44
∴x≤4
Because there are five nonnegative integers not greater than 4, namely 0, 1, 2, 3, 4, the nonnegative integer solution of inequality 10(x+4)+x≤84 is 4, 3, 2, 1 0.
Example 5 Solving Inequalities about X
( 1)ax+2≤bx- 1(2)m(m-x)> n(n-x)
Analysis: The methods and steps to solve the letter coefficient inequality are similar to those to solve the number coefficient inequality, but the letter coefficient is often discussed in the process of solving, which increases the difficulty of the topic. This kind of problem mainly examines the ability to analyze and classify the problem: not only do you need to know when to discuss the classification, but you also need to be able to discuss the classification accurately (which will be explained with the solution of an example).
Solution: (1)∵ax+2≤bx- 1
∴ax-bx≤- 1-2
That is, (a-b)x≤-3
At this time, the form of inequality solution should be obtained according to the different values of X letter coefficient.
That is, (n-m) x > N2-m2
When m > n and n-m < 0, ∴ x < n+m;
When m < n, n-m > 0, ∴ x > n+m;
When m=n, n-m=0, n2=m2, n2-m2=0, the original inequality has no solution. This is because the values on both sides of the inequality are zero and can only be equal, so the inequality does not hold.
Example 6 Solving Inequalities about X
3(a+ 1)x+3a≥2ax+3。
Analysis: Because X is an unknown number, A is regarded as a known number, and A can be any rational number, so when applying the same solution principle, we should distinguish the situations and deal with them separately.
Solution: Remove the brackets and get
3ax+3x+3a≥2ax+3
Move items, get
3ax+3x-2ax≥3-3a
Merge similar projects to obtain
(a+3)x≥3-3a
(3) When a+3=0, that is, a=-3, 0 x ≥ 12 is obtained.
There is no solution to this inequality.
Note: When dealing with letter coefficient inequality, we must first find out which letter is unknown and regard other letters as known numbers. When using the same solution principle to change the unknown coefficient into 1, we should make reasonable classification and discuss them one by one.
Example 7 When m is what value, the solution of equation 3 (2x-3m)-2 (x+4m) = 4 (5-x) about x is not positive.
Analysis: according to the meaning of the question, first solve the equation by taking m as a known number, then list the inequality about m according to the conditions of the solution, and then solve this inequality to find the value or range of m. Note that "non-positive number" is a number less than or equal to zero.
Solution: The known equation is 6x-9m-2x-8m=20-4x.
The solution can be 8x=20+ 17m.
It is known that the solution of the equation is non-positive, so
Example 8 If the solution of equation 5x-(4k- 1)=7x+4k-3 about x is: (1) non-negative and (2) negative, try to determine the value range of k. 。
Analysis: To determine the value range of k, we should regard k as a known number, and work out the solution X of the equation according to the steps of solving a linear equation with one variable (expressed by the algebraic expression of k). At this time, according to whether the solution of the equation is negative or not, we can get the inequality about K, and then we can get the range of K. What needs to be emphasized here is that this problem does not directly solve inequalities, but obtains inequalities according to known conditions, which belongs to the application of inequalities.
Solution: The known equation is 5x-4k+ 1=7x+4k-3.
Solvable -2x=8k-4.
That is, x=2( 1-2k)
(1) It is known that the solution of the equation is non-negative, so
(2) It is known that the solution of the equation is negative, so
Example 9 When the value of x is in what range, the algebraic expression -3x+5:
(1) is a negative number (2) is greater than -4.
(3) Less than -2x+3 (4) Not more than 4x-9
Analysis: The key to solving the problem is to accurately translate written languages such as "negative number", "greater than", "less than" and "not greater than" into digital symbols.
Solution: (1) According to the meaning of the question, it should be inequality.
Solution set of -3x+5 < 0
In order to solve this inequality, you must
(2) According to the meaning of the question, the inequality should be found.
Solution set of -3x+5 >-4
In order to solve this inequality, you must
x 0。
It is complicated to remove brackets directly. Note that all terms on the left contain the factor x-3, which can be solved quickly by using the inverse distribution law.
Solve the original inequality into
(x-3)(278-35 1×2+463)>0，
That is, 39 (x-3) > 0, so x > 3.
8. Clever use of overall merger
Example 9 Solving Inequalities
3 { 2x- 1-[3(2x- 1)+3]} > 5。
Take 2x- 1 as a whole, remove the brackets to get 3 (2x-1)-9 (2x-1)-9 > 5, and combine the whole to get -6 (2x- 1) > 14.
9. Clever disassembly
Example 10 Solving Inequalities
It is considered that dividing -3 into three negative 1 and then combining with the other three items can skillfully solve this problem.
Solving the original inequality is transformed into
X- 1≥0, so X ≥ 1.