solution
eliminate
1) into the elimination method.
The general steps of substitution elimination method are:
1. Choose an equation with relatively simple coefficients and convert it into the form of y = ax +b or x = ay+b;
2. Substitute y = ax+b or x = ay+b into another equation to eliminate an unknown number, thus changing another equation into a linear equation;
3. Solve this one-dimensional linear equation and find the value of x or y;
4. Substitute the calculated value of x or y into any equation (y = ax +b or x = ay+b) to find another unknown;
5。 The values of two unknowns are connected by braces, which is the solution of binary linear equation.
Example: solving equation: x+y=5①
6x+ 13y=89②
Solution: x=5-y③ from ①.
Substituting ③ into ② gives 6(5-y)+ 13y=89.
Get y=59/7.
Substitute y=59/7 into ③ to get x=5-59/7.
Get x=-24/7.
∴ x=-24/7
Y=59/7 is the solution of the equation.
We call this method of eliminating the unknown quantity by "replacement method" to find the solution of the equation "replacement method" for short.
2) Addition, subtraction and elimination methods
(1) In binary linear equations, if there is the same unknown with the same coefficient (or inverse number), you can directly subtract (or add) an unknown;
(2) In the binary linear equations, if the situation in (1) does not exist, we can choose an appropriate number to multiply the two sides of the equation, so that the coefficient of one of the unknowns is the same (or vice versa), and then subtract (or add) the two sides of the equation respectively to eliminate an unknown, and get a linear equation;
(3) solving this one-dimensional linear equation;
(4) Substituting the solution of the obtained one-dimensional linear equation into the equation with relatively simple coefficients of the original equation set to find the value of another unknown;
⑤ Connecting the values of two unknowns with braces is the solution of binary linear equations.
The first method of solving equations by addition, subtraction and elimination
Example: Solving Equation:
x+y=9①
x-y=5②
Solution: ①+②
De: 2x= 14
∴x=7
Substitute x=7 into ①.
D: 7+y=9
∴y=2
The solution of the equations is: x=7.
y=2
The second method of adding, subtracting and eliminating equations.
Example: Solving Equation:
x+y=9①
x-y=5②
Solution: ①+②
De: 2x= 14
∴x=7
①-②
D: 2y=4
∴y=2
The solution of the equations is: x=7.
y=2
By using the properties of equations, the absolute value of the coefficient in front of an unknown in two equations in a group of equations is equal, and then the two equations are added (or subtracted) to eliminate the unknown, so that the equation can be solved with only one unknown, and then it is substituted into one equation in this group of equations. Such a method for solving binary linear equations is called addition and subtraction, or addition and subtraction for short.
3) sequential exclusion method
Let the binary linear equation be:
ax+by=c ( 1)
dx+ey=f (2)
(A, B, D and E are the coefficients of X and Y)
If: a≠0, equation (3) is obtained:
In Formula (3), we can get the formula for solving the binary linear equations:
The above process is called "sequential elimination method", and the principle of solving multivariate equations is the same.
Alternative method
Example 2, (x+5)+(y-4)=8.
(x+5)-(y-4)=4
Let x+5 = m and y-4 = n.
The original equation can be written as
m+n=8
m-n=4
The solution is m = 6 and n = 2.
So x+5 = 6 and y-4 = 2.
So x= 1, y=6.
Features: The two equations contain the same algebraic expressions, such as x+5 and y-4 in the title, and the simplification of the equations after substitution is also the main reason.
Parameter setting method
Example 3, x:y= 1:4
5x+6y=29
Let x = t and y = 4t.
Equation 2 can be written as 5t+6*4t=29.
29t=29
t= 1
So x= 1, y=4.