Related definitions
Binary linear equations
1. Definition
If an equation contains two unknowns, and the degree of the unknowns is 1, such an integral equation is called a binary linear equation.
The value of the two unknowns that make the values on both sides of a linear equation in two variables equal is called the solution of a linear equation in two variables.
2. General form
ax+by+c=O(a, b≠0).
3. Solving method
Using the divisible characteristics of numbers combined with the method of substitution and elimination to solve. (The mantissa characteristic of the number can be used, and the parity of the number can also be used.)
Binary linear equations
1. Definition
An equation system consisting of two linear equations and containing two unknowns is called a binary linear equation system.
Generally, the common solution of two binary linear equations of a binary linear equation system is called the solution of the binary linear equation system.
2. General form
(where a1, a2, b1, b2 are not zero at the same time)
3. Solving method
elimination method, substitution method, parameter setting method, image method, solution vector method.
Solution
Elimination Method
1) Substitution Elimination Method
The general steps of substitution elimination method are: p>
1. Select an equation with a relatively simple coefficient to transform it into the form y = ax + b or x = ay + b;
2. Change y = ax + b or x = ay + b is substituted into another equation to eliminate an unknown number, thereby turning the other equation into a one-dimensional linear equation;
3. Solve this one-variable linear equation and find the value of x or y;
4. Substitute the calculated x or y value into any one of the equations (y = ax + b or x = ay + b) to find another unknown number;
5 . Put the values of the two unknowns together in braces, and this is the solution of the linear equation in two unknowns.
Example: Solving equations: x+y=5①
6x+13y=89②
Solution: x=5-y③
>Substituting ③ into ②, we get 6(5-y)+13y=89
get y=59/7
Substitute y=59/7 into ③, get x=5-59/7
get x=-24/7
∴ x=-24/7
y=59/7 is the system of equations The solution of
We call this method of eliminating an unknown number through "substitution" to find the solution of the system of equations called elimination by substitution (elimination by substitution), or substitution for short.
2) Addition, subtraction and elimination method
①In the binary linear equations, if the coefficients of the same unknown number are the same (or opposite to each other), they can be directly compared Subtract (or add) to eliminate an unknown number;
②In the binary linear equations, if the situation in ① does not exist, you can choose an appropriate number to multiply the two sides of the equation to make one The coefficients of the unknowns are the same (or opposite to each other), and then subtract (or add) the two sides of the equation respectively to eliminate an unknown number to obtain a linear equation in one variable;
③Solve this linear equation in one variable;
p>
④Substitute the solution of the obtained one-variable linear equation into the equation with the simpler coefficients of the original equations, and find the value of another unknown;
⑤Use the two obtained unknowns with the values The braces are joined together, and this is the solution of the system of linear equations in two unknowns.
The first method of solving equations by adding, subtracting and eliminating elements
Example: Solving equations:
x+y=9①
xy=5②
Solution: ①+②
Get: 2x=14
∴x=7
Substitute x=7 into ①
to get: 7+y=9
∴y=2
∴The solution of the equations is: x=7
y=2
The second method of solving equations by adding, subtracting and eliminating elements
Example: Solving equations:
x +y=9①
xy=5②
Solution: ①+②
Get: 2x=14
∴x=7< /p>
①-②
Get: 2y=4
∴y=2
The solution of the system of equations is: x=7< /p>
y=2
Using the properties of the equation to make the absolute value of the coefficient before an unknown number in the two equations in the system of equations equal, and then adding the two equations (or Subtract) to eliminate this unknown, so that the equation contains only one unknown and be solved, and then substitute it into one of the equations. This method of solving binary linear equations is called elimination by addition-subtraction, abbreviated as addition-subtraction.
3) Sequential elimination method
Suppose the binary linear equation system is:
ax+by=c (1)
dx+ey=f (2)
(a,b,d,e are the coefficients of x,y)
If:, then the formula (3):
If (3) in the formula,
The formula for solving the two-variable linear equations can be obtained:
The above process is called "Sequential Elimination Method". For multi-element systems, the solving principle is the same.
It should be that there are only operations between numbers in the process of solving, and there is no operation of the entire formula, so this method is widely used in computers.
Substitution method
Example 2, (x+5)+(y-4)=8
(x+5)-(y-4 )=4
Let x+5=m,y-4=n
The original equation can be written as
m+n=8
mn=4
Solve m=6,n=2
So x+5=6,y-4=2
So x =1,y=6
Characteristics: Both equations contain the same algebraic formula, such as x+5, y-4 in the question, the main reason is that the equation can be simplified after changing the element.
Parameter setting method
Example 3, x:y=1:4
5x+6y=29
let x=t ,y=4t
Equation 2 can be written as: 5t+6*4t=29
29t=29
t=1
So x=1,y=4
The derivation process of binary linear equations:
< p>
There is only one y unknown in the last formula, find the y value (y=?), and then substitute a1x+b1y=k1; Find X.
Example question:
y=(2-3/4*0)/(1-3/4*2)=2/(-1 /2)=-4
3x-4=2 or 4x-8=0 x=2
Deduction of a simple equation:
< p>Equation=0; unknown number 0; 1
Image method
The binary linear equations can also be used as images The method is to rewrite the corresponding binary linear equation into a linear function expression to draw an image in the same coordinate system, and the coordinates of the intersection of the two straight lines are the solution of the binary linear equation system.
Vector solution method
Now there is a two-variable linear equation~~~①
Set up a matrix, a vector sum, and define it according to the product of a matrix and a vector. Comparing the equations again, we can see that there are the following relationships:
~~~②
We call ② the matrix form of the equations ①
and the matrix A can be seen Doing is a linear transformation p, that is, the vector is obtained after the vector is transformed according to the linear transformation p. Therefore, the process of solving the equation can be regarded as looking for a vector, which is obtained by linear transformation p. Because this is the process of finding a vector, it can also be called a solution vector.
Understand the above sentence intuitively. For example, if you rotate a vector a counterclockwise by 30° to get a new vector b, then you can get a by rotating b clockwise by 30°. For another example, if the horizontal and vertical coordinates of a vector a are expanded by n times to obtain the vector b, then after the horizontal and vertical coordinates of b are reduced by n times, a must be obtained. Therefore, when b and the linear transformation relationship are known, the a obtained is the solution of the equation.
The linear transformations of matrix A and its inverse matrix are mutually inverse, so the process of solving the vector is equivalent to finding the inverse matrix of the matrix. According to the properties of the matrix, the necessary and sufficient condition for a matrix to have an inverse matrix is the second-order determinant=ad-bc≠0. Therefore, the necessary and sufficient condition for a system of equations to be solved is ad-bc≠0.
According to the method of finding the inverse matrix, the inverse matrix of is
∴ = =
That is, the solution of the equation system is
This method It can also be used as the root-finding formula of a binary linear equation system. (The premise is ad-bc≠0!)
Example question: Solve a system of linear equations in two unknowns by the solution vector method
In this question, a=3, b=1, c= 4, d=2, e=2, f=0, ad-bc=3*2-1*4=2≠0
∴The system of equations has a solution, the solution is
x=(de-bf)/(ad-bc)=(2*2-1*0)/2=2
y=(af-ce)/(ad-bc)=( 3*0-4*2)/2=-4
Three types of solutions
Generally, the left and right values of the two equations The value of two equal unknowns is called the solution of a system of linear equations in two unknowns. The process of finding solutions to a system of equations is called solving a system of equations. Generally speaking, a binary linear equation has an infinite number of solutions, and the solution of a binary linear equation system has the following three situations:
unique solution
such as the equation system x+y =5①
6x+13y=89②
x=-24/7
y=59/7 is the solution of the system of equations
Is there an array solution
For example, the equation system x+y=6①
2x+2y=12②
Because these two equations are actually one equation ( It is also called "the equation has two equal real roots"), so there is no array solution for this type of equation system.
Another example: x+(yx)=y①
y+(xy)=x②
No solution
Such as the equation system x+ y=4①
2x+2y=10②,
Because the equation ② is simplified to
x+y=5
this Contradicts with equation ①, so this type of equation system has no solution.
The solution of a system of linear equations with two variables can be judged by the ratio of coefficients, such as the following system of linear equations with two variables for x and y:
ax+by=c< /p>
dx+ey=f
When a/d≠b/e, the system of equations has a set of solutions.
When a/d=b/e=c/f, whether the equation system has an array solution.
When a/d=b/e≠c/f, the system of equations has no solution.
Differences
Differences from quadratic equations in one variable
1. Definition and general form:
2. Solution: ⑴Direct leveling method (note the characteristics)
⑵Matching method (note the steps—reject the root formula)
⑶Formula method:
⑷Factor Decomposition method (feature: left=0)
3. The discriminant of the root:
4. The relationship between the root and the top of the coefficient:
Inverse theorem: If, then the quadratic equation of one variable with the root is:.
5. Commonly used equations:
⑵Basic idea:
⑶Basic solution:
①Multiplication method (pay attention to skills!!)
② Exchange yuan method (example, )
XY=Y-1
example
1. A reservoir plans to send water to both places A and B, and land A needs The water is 1.8 million cubic meters, and the B place needs 1.2 million cubic meters of water. Now it has been sent twice, the first time to send water to A for 3 days, and B to send water for 2 days, a total of 840,000 cubic meters of water was sent; the second time to A Water was sent to the ground for 2 days, and to the ground for 3 days, a total of 810,000 cubic meters of water was sent. If the water is delivered at this schedule, ask: How many days will it take to complete the task of delivering water to A and B?
Suppose: the speeds of staying A and B sending water are X and Y respectively
3X+2Y=84
2X+3Y=81, we get X=18 Y=15
A place needs 180/18-5=5 days and B place another 120/15-5=3 days
2. A student asks the teacher: "You How old is this year?" The teacher said wittily, "You were only born when I was your age, and when you were my age, I was 37 years old." How old are the teacher and the student?
Suppose: the ages of the teacher and the students are X, Y years old
X+XY=37 can be solved by X=25 Y=13
Others
Note
The binary linear equations are not necessarily composed of two binary linear equations together! More than just one kind.
It can also be composed of one or more binary linear equations alone.
Key points: One-dimensional linear equations, one-variable quadratic equations, and two-variable linear equations; related application problems of equations (especially travel, engineering problems)
Based on the nature of the equation
1. a=b←→a+c=b+c
2. a=b←→ac=bc (c>0)
A list of equations (sets) to solve applied problems
An overview
A list of equations (sets) solutions Application problems are an important aspect of linking mathematics with practice in middle school. The specific steps are:
⑴ Review questions. Understand the meaning of the question. Find out what is the known quantity in the problem, what is the unknown quantity, and what is the equivalence relation given and involved in the problem.
⑵Set Yuan (unknown). ①Direct unknown ②Indirect unknown (often used both). Generally speaking, the more unknowns, the easier the equation is to list, but the more difficult it is to solve.
(3) Use algebraic expressions containing unknown numbers to express the relevant quantities.
⑷Find the equivalence relationship (some are given by the title, and some are given by the equivalence relationship involved in the problem), and formulate equations. Generally, the number of unknowns is the same as the number of equations.
⑸Solve equations and test.
⑹Answer.
To sum up, the essence of formulating equations (groups) to solve applied problems is to first transform actual problems into mathematical problems (set elements, formulae equations), and solve actual problems caused by the solution of mathematical problems (List the equation and write the answer). In this process, the equations play a role in linking the past and the future. Therefore, formulating equations is the key to solving applied problems.
Two commonly used equal relations
1. Travel problem (uniform speed movement)
Basic relationship: s=vt
⑴ Encounter problem (start at the same time);
⑵ Track down the problem (start at the same time); < /p>
If A starts t hours later, then B starts, and then catches up with A at B, then
3 sail in the water;
2. Ingredients: solute=solution×concentration
solution=solute+solvent
3. Growth rate problem
Growth rate=value after growth/value before growth
4. Engineering issues
Basic relationship: workload = work efficiency × working time (the workload is often regarded as the unit "1").
5. Geometry problems
The Pythagorean theorem is commonly used, the area and volume formulas of geometric bodies, similar shapes and related proportional properties, etc.
Three pay attention to the mutualization of language and analytical formula:
For example, "more", "less", "increased", "increased to (to)", "simultaneously" , "Expanded to (to)", "Expanded",...
Another example, a three-digit number, the hundreds digit is a, the tens digit is b, and the ones digit is c, then This three-digit number is: 100a+10b+c, not abc.
Four attention to write the equality relationship from the language description:
For example, x is greater than y by 3, then x-y=3 or x=y+3 or x-3=y. For another example, if the difference between x and y is 3, then x-y=3. Five pay attention to unit conversion
For example, the conversion of "hour" and "minute"; the unit of s, v, t is consistent, etc.
Using program to solve
The binary linear equations can be solved by the computer program by the sequential elimination method. The following is an example written in C++:
#include using namespace std ;class EYYCFCZ{public: void get(double a00, double a01, double a10, double a11, double b0, double b1); double returny(); double return x() {x = (b[0]-a[0] [1] * y) / a[0][0]; return x; }private: double a[2][2]; double b[2]; double x,y;};double EYYCFCZ::returny() {double m = a[1][0] / a[0][0]; double a_11 = a[1][1]-m*a[0][1]; double b_1 = b[1]-m *b[0]; y = b_1 / a_11; return y;}void main(){ double a00, a01, a10, a11, b0, b1; cout <> a00; cout <> a01; cout < ;> b0; cout <> a10; cout <> a11; cout <> b1; EYYCFCZ fc; fc.get(a00, a01, a10, a11, b0, b1); double y = fc .returny(); double x = fc.returnx(); cout <