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There is a section of uphill road and a section of level road from point a to point B. if the uphill road is kept at 3km per hour, the level road is 4km per hour, and the downhill road is 5km per hour, then it takes 54 minutes to get from point a to place B, and 42 minutes to place a to place A. how much is the whole journey from point a to point B?
Suppose that the uphill road from a to B is XKM, and the flat road is YKM,
From the meaning of the title:
X
3+x
4＝54
Sixty
Y
4+x
5＝42
Sixty
The solution is as follows:
x＝1.5
y＝1.6
So: x + y = 3.1km
A: the whole journey from a to B is 3.1 kilometers
1. If two groups of solutions of a bivariate linear equation are respectively {x = - 4 {x = 5. Try to find the bivariate first order equation ｛y=-4｛y=2 2. There are whole bags of grain in warehouse A and warehouse B. If 90 bags of grain are transferred from warehouse A to warehouse B, the stock of warehouse B is twice as much as that of warehouse A. if several bags are transferred from warehouse B to warehouse A, the grain in warehouse A is 6 times as much as that in warehouse B. how many bags of grain should be stored in warehouse A at least? (detailed explanation) 3. In order to celebrate the arrival of Beijing Olympic Games, the garden department decided to use the existing 3250 pots of a flower and B flower to match a certain gardening style, a total of 50 pots were placed on both sides of Jinshan Avenue. It is known that 50 pots of a flower and 80 pots of B flower are needed to match this shape. How many pots of a flower and B flower are there in the garden department 1. The two solutions are x = - 4, y = - 4; X = 5, y = 2
1. Let this equation be ax + by + C = 0, substitute two groups of solutions into the equation with - 4a-4b + C = 0, 5A + 2B + C = 0,2 and subtract to get 9A + 6B = 0
The solution of two variables quadratic equation Seeking the law,
Two variable quadratic equation y = ax ^ 2 + BX + C I summarize the following solutions
When a, B, C are not equal to 0, there is the following solution
First consider the cross phase method, if you can not consider the formula method or formula method, such as;
X ^ 2 + X-6 = 0
X ^ 2 + X-5 = 0 instead of using the cross matching method, let's do the matching method first
x^2+x+(1/2)^2-(1/2)^2-5=0
(x+1/2)^2-2¼=0
(x + 1 / 2) = positive and negative √ 21 / 2
= - 2 / 2 / 2 - √
Using formula method
X = - B + √ (b ^ 2-4ac) / 2A or x = - B - √ (b ^ 2-4ac) / 2A
Because a = 1, B = 1, C = - 5 can be directly substituted into two values
When B is equal to 0, we can cut the square as follows
3x^2-27=0
3x^2=27
x^2=9
X = - 3 or x = 3
When C is equal to 0, use the common factor method, such as
4x^2+6x=0
2x(2x+3)=0
So x = 0 or x = - 3 / 2
The solution of the following bivariate quadratic equation 1000Z-YZ=17100 24.1Y-YZ=7000 It's better to have steps. Thank you
Z(1000-Y)=17100 (1)
Y(24.1-Z)=7000 (2)
From (1): z = 17100 / (1000-y) (3)
Substitute (2): y [(24.1-17100 / (1000-y)] = 7000
Finishing: y [24.1 (1000-y) - 17100] = 7000 (1000-y)
Y(24100-24.1Y-17100)=7*10^6-7000Y
24100Y-24.1Y^2-17100Y=7*10^6-7000Y
24.1Y^2 - 14000Y + 7*10^6 = 0
The solution is: y = 290.4564 ± 0.4539 I (complex number)
Substituting into (3), the solution: z = (omitted)
The solution of bivariate equation of first degree (I don't know much about it)
Substitution and elimination
(1) Concept: a certain unknown number of an equation in the system of equations is expressed by an algebraic expression containing another unknown number, which is substituted into another equation and eliminated by one unknown number to obtain a one variable linear equation, and finally the solution of the equation system is obtained. This method of solving the equation system is called substitution elimination method, abbreviated as substitution method. [4]
(2) The steps of solving the system of binary first order equations by substitution method
① In this paper, we select a variant of a quadratic equation with simple coefficients, and express another unknown number with an algebraic expression containing one unknown number;
② The transformed equation is substituted into another equation, and an unknown number is eliminated to obtain a one variable linear equation (in the process of substitution, it should be noted that it can not be substituted into the original equation, but can only be substituted into another equation without deformation, so as to achieve the purpose of elimination);
③ By solving the equation of degree of one variable, the value of the unknown number can be obtained;
④ The value of the unknown number obtained is substituted into the transformed equation in (1) to obtain the value of another unknown number;
⑤ The value of two unknowns is the solution of the system of equations;
⑥ Finally, test (replace the original equations to test whether the equation satisfies the left = right)
Example:
{x-y=3 ①
{3x-8y=4②
From ①, x = y + 3 ③
③ We can get
3（y+3)-8y=4
Y=1
Bring y = 1 into 3
X = 4
Then: the solution of the system of bivariate linear equations
{x=4
{y=1
Addition, subtraction and elimination
(1) Concept: when the coefficients of an unknown number of two equations are equal or opposite to each other, the unknown number is eliminated by adding or subtracting the two sides of the two equations, so as to change the binary linear equation into the one variable one degree equation, and finally obtain the solution of the equation system. This method of solving the system is called the addition and subtraction elimination method, referred to as the addition and subtraction method
(2) The steps of adding and subtracting to solve the system of quadratic equations of two variables
① By using the basic properties of the equation, the coefficient of an unknown number in the original equation system is transformed into the form of equal or opposite number;
② Then use the basic properties of the equation to add or subtract the two equations after deformation, eliminate an unknown number, and get a one variable linear equation (be sure to multiply both sides of the equation by the same number, do not multiply only one side, and then use subtraction if the coefficients of unknowns are equal, or add if the coefficients of unknowns are opposite to each other);
③ By solving the equation of degree of one variable, the value of the unknown number can be obtained;
④ Substituting the value of the unknown into any equation of the original equations, the value of another unknown number can be obtained;
⑤ The value of two unknowns is the solution of the system of equations;
⑥ Finally, check whether the result is correct (put it into the original equation group to test, whether the equation meets the left = right)
For example:
{5x+3y=9①
{10x+5y=12②
Enlarge 1 by 2 times to get 3
10x+6y=18
③ - 2
10x+6y-(10x+5y)=18-12
Y=6
Then bring y = into ①. ② or ③
The solution is: {x = - 1.8
{y=6
The solution of bivariate linear equation needs an example, which must be more than 5
Question: 3x + 4Y = 11
5x+2y=9 ②
② * 2
10x+4y=18 ③
③ - 1
（10x+4y）-（3x+4y）=18-11
7x=7
X=1
Put x = 1 into (2) to get
5*1+2y=9
2y=4
Y=2
∴x=1
Y=2
The solution of bivariate linear equation It's better to give a more difficult example
The general form of bivariate linear equation
Ax ^ 2 + BX + C = 0 has formula solution
The discriminant delta = B ^ 2-4ac
When delta > 0, there are two different real roots X1 = (- B + root (delta)) / 2A; x2 = (- b-radical (delta)) / 2A
When delta = 0, there are two identical real roots X1 = x2 = - B / 2A
When delta
The solution of the first order equation of two variables, by the way
If there are two unknowns in the equation system, and the degree of the unknown term is 1, such equation is called bivariate first order equation. Two bivariate linear equations together constitute a binary linear equation system
The solution and analysis of solving the problem of the first degree equation of two variables A three digit number is five times as much as a two digit number. If the three digit number is placed on the left of the two digit number, it is 18648 smaller than the five digit number on the right?
Let three digits be x and two digits y, then x = 5Y (1)
Three digits are placed to the left of two digits, which is the sum of three digits and two digits
Three digits are placed to the right of two digits, which is 1000 times the sum of two and three digits
It is known that 1000y + X - (100x + y) = 18648 (2)
Solution (1) (2)
X = 185, y = 37
We will pay RMB 1.5 yuan, 10 yuan, 20 yuan, 50 yuan and 100 yuan in our country to pay RMB 2 yuan, 3 yuan, 999 yuan and 1000 yuan, What kind of currency at least
Because the maximum payment is 1000 yuan, the sum of all denominations should be 1000 yuan. To achieve the minimum, there must be 99 pieces of 100 yuan. The remaining 100 should be arranged and combined to form 1,5,10,20,50. Therefore, at least one 50, one 20, one 10, one 5, five 1 is needed. Therefore, the answer is: 5 pieces of 1 yuan, 1 piece of 5
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