Ternary linear equations

Concept

Containing three identical unknowns, the degree of the term containing the unknown in each equation is once, which is called a ternary linear equation system. In the system of equations, if there are less than 3 equations, it is impossible to find the solution of all the unknowns. Therefore, the general ternary linear equation is a system of equations composed of three equations.

Solution

The basic idea of ​​solving ternary linear equations is to eliminate the element through "substitution" or "addition and subtraction", turning the "ternary" into "binary" , So that the solution of the ternary linear equations is transformed into the solution of the binary linear equations, and then into the solution of the linear equations of one variable.

Their main solution methods are addition and subtraction elimination method and substitution elimination method. They usually use addition and subtraction elimination method. If the equation is difficult to solve, use the substitution elimination method, which varies depending on the problem. The idea is to use the elimination method to gradually eliminate the element.

Steps: ①Using substitution or addition and subtraction, eliminate an unknown number, and get a system of linear equations in two variables;

②Solve the system of linear equations in two variables, and get two The value of an unknown number;

③Substitute the values ​​of these two unknowns into a simpler equation in the original equation, find the value of the third unknown, and write these three numbers together. The solution to the system of linear equations in three variables.

Learning objectives and requirements

1. Understand the concept of ternary linear equations; be proficient in the solution of simple ternary linear equations; be able to choose simple solutions to special Three-dimensional linear equations.

2. Can solve simple ternary linear equations by using substitution elimination method, addition and subtraction elimination method, and choose a reasonable and simple method to solve the equation set, and cultivate computing ability.

3. Through the observation and analysis of the characteristics of the unknown coefficients in the equations, it is clear that the main idea of ​​the solution of the ternary linear equations is "elimination", so as to promote the transformation, cultivation and development of the unknown into the known Logical thinking ability.

4. Able to transform a three-element linear equation system into a two element linear equation system through elimination, and then eliminate the element to transform into a one element linear equation and transform some algebraic problems into equation system problems, and initially apply transformation ideas To solve problems and develop thinking skills.

Application

Simple application of ternary linear equation:

1.

Solve x, y, z value.

Solution: ①+②×2 gets: 5x+7z=21 ④

②+③ gets: x+z=5 ⑤

Lianli④ , ⑤Get:

Using the binary linear equation solution to get:

Set x=7, z=-2 Substituting ①, it can be solved to get y=1

So the solution of the original system of equations is:

Complex application of ternary linear equations:

a2x+b2y+c2z=d2

a3x+b3y+c3z=d3 }Group:

x, y, z are unknown, a1, a2, a3, b1, b2, b3, c1, c2, c3, d1, d2, d3 are constants, and solve the x, y, z values.

{ a1x+b1y+c1z=d1 ① a2x+b2y+c2z=d2　②a3x+b3y+c3z=d3 ③ }

Solution: {b1y=d1-a1x-c1z ④

b2y=d2-a2x-c2z ⑤

b3y=d3-a3x-c3z ⑥}

④÷⑤

b1 /b2*(d2-a2x-c2z)=d1-a1x-c1z ⑦⑤÷⑥b2/b3*(d3-a3x-c3z)=d2-a2x-c2z ⑧

Get from ⑦: b1 /b2*d2-b1/b2*a2x-b1/b2*c2z=d1-a1x-c1z

a1x-b1/b2*a2x+c1z-b1/b2*c2z=d1-b1/b2 *d2

(a1-b1/b2*a2)x+(c1-b1/b2*c2)z=d1-b1/b2*d2

(c1-b1/b2 *c2)z=d1-b1/b2*d2-(a1-b1/b2*a2)x ⑨

Get from ⑧:

b2/b3*d3-b2/ b3*a3x-b2/b3*c3z=d2-a2x-c2z

a2x+c2z-b2/b3*a3x-b2/b3*c3z=d2-b2/b3*d3

(c2-b2/b3*c3)Z=d2- b2/b3*d3-(a2-b2/b3*a3)x ⑩

⑨÷⑩

[(c1-b1/b2*c2)÷(c2-b2/ b3*c3)]*[d2-b2/b3*d3-(a2-b2/b3*a3)x]=d1-b1/b2*d2-(a1-b1/b2*a2)x ⑾

In ⑾, a1, a2, a3, b1, b2, b3, c1, c2, c3, d1, d2, d3 are all constants, and only X is an unknown number, so the value of X has been solved. Substitute the constant into the formula to find the value of X, then substitute the value of X into ⑨ or ⑩ to find the value of Z, and then substitute the value of X　Z into one of the original formulas ①②③ to find the value of y.

The three unknown values ​​of x, y and z in the ternary linear equation have been solved.