Under what circumstances is a system of binary equations unsolved

That is, when the slope of the corresponding straight line is the same, there may be countless solutions or no solutions.

For example, y=kx+b

y=nkx+c

n is any non-zero number, and if b=nc, then the two equations are equivalent, with an infinite number of solutions; But if b is not equal to nc, then there is no solution to the equation.

I think the situation of no solution and countless solutions is.

1.Define fields that are not allowed, such as y=log10(x) and x=-sqrt(y)2The slope of the two equations is the same but the relationship is different at the intersection of the y-axis, e.g. y=2x+3 and y=2x+6

In fact, the unsolved binary equation is equivalent to two direct in the coordinate system without an intersection point, so how to arrange the two straight lines so that they have no intersection point? (Note, the straight line is infinitely extended) the landlord wants to go down, use his brain, and don't think of asking me again.

I'd like to ask the upstairs guys: Is there an infinite number of solutions to two parallel lines???

a1x+b1y=c1

a2x+b2y=c2

When a1 a2=b1 b2=c1 c2 there are an infinite number of solutions.

When a1 a2=b1 b2 is not equal to c1 c2 there is no solution.

As long as a and b are not 0 at the same time.

As you can see by drawing a diagram, two straight lines are parallel (each equation represents a straight line): an infinite number of solutions.

There is an intersection: there is a solution.

The solution of a system of binary linear equations is the value of the unknowns that make the left and right sides of the equation equal.

The understanding of the solution of a binary linear equation should pay attention to the following:

In general, there are an infinite number of solutions to a binary equation, and each solution refers to a pair of values, not to the value of a single unknown.

A solution to a binary equation is the value of a pair of unknowns that equalizes the left and right sides of the equation; Conversely, if a set of values equalizes the left and right sides of a binary equation, then that set of values is the solution of the equation.

When finding the solution of a binary linear equation, the usual practice is to use an unknown to represent another unknown, and then give a value to this unknown, and accordingly obtain the value of another unknown, so that a solution of the binary linear equation can be obtained.

Steps to solve a system of binary equations by addition and subtraction:

Using the basic properties of equations, the coefficients of an unknown number in the original system of equations are reduced to equal or opposite numbers.

Then use the basic properties of the equation to add or subtract the two deformed equations, eliminate an unknown number, and obtain a unary 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 unknown coefficients are equal, and add if the unknown coefficients are opposite to each other).

Solve this unary equation and find the value of the unknown.

Substituting the value of the obtained unknown into any one of the equations in the original system of equations to find the value of another unknown.

Three cases of solutions to binary systems of linear equations a1x+b1y+c1=0, a2x+b2y+c2=0:

1) When a1 a2≠b1 b2, the system of equations has a unique solution, (2) when a1 a2=b1 b2≠c1 c2, the system has no solution, (3) when a1 a2=b1 b2=c1 c2, the system of equations has an infinite number of solutions.

Binary systems of equations can be formed into the following patterns:

y=ax+b

y=ax+b

a, b, a, b are real numbers and a≠0, a≠0).

If you take the two equations into the above pattern and find that (a=a and b≠b), then the system of equations has no solution.

In junior high school knowledge and above, the use of plane analytic geometry can easily explain the above situation:

In Cartesian coordinates, each binary linear equation can be expressed as a straight line on a plane, and the two equations in a system of binary linear equations are two lines, and the solution of the system of equations is the intersection of these two straight lines.

The absence of a solution to a system of equations is described in Cartesian coordinates as two lines that do not intersect, i.e., two lines are parallel.

The sufficient and necessary conditions for two straight lines to be parallel are:

The angle is the same (a=a) and the intercept is different (b≠b).

The conditions for the unsolvable system of binary linear equations are as follows:

1、y=ax+b

2、y=ax+b

The elimination of the first a, b, a, b are real numbers and a≠0, a≠0).

If we find that (a=a and b≠b) after the two equations are reduced to the above model, then the system of equations has no solution.

Introduction to Binary Linear Equations:

1. Definitions. If an equation contains two unknowns, and the number of unknowns is 1, such an integral equation is called a binary linear equation.

The values of two unknowns that equal the values on both sides of a binary equation are called the solutions of the binary equations.

2. General form.

ax+by+c=0（a，b≠0）。

3. Solving method.

The divisibility of the number is used in combination with the method of substitution and exclusion to solve the problem. (You can take advantage of the mantissa feature of numbers, and you can also use the parity of numbers.) ）

How to solve a binary equation:

Addition, subtraction, and subtraction:

1. In the binary system of linear equations, if there are the same coefficients of the same unknown (or opposite to each other), it can be directly subtracted (or added) to eliminate an unknown;

2. In the system of binary linear equations, if there is no such situation, you can choose an appropriate number to multiply both sides of the equation so that the coefficients of one of the unknown numbers 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 unary equation;

3. Solve this unary equation;

4. Substitute the solution of the one-dimensional one-dimensional equation into the equation with relatively simple coefficients of the original equation, and find the value of another unknown;

5. The values of the two unknowns obtained are connected in curly brackets, which is the solution of the binary system of equations.

Let this system of equations be.

ax+by=c

dx+ey=f

When a d=b e=c f, there are an infinite number of solutions.

When a d=b e≠c f, there is no solution.

When a d≠b e, there is a unique solution.

When does a system of binary linear equations have an infinite number of solutions, and when does it have no solutions?

a 2 = 3 1 is not equal to = 9 b

That is, when a = 6 and b is not equal to 3, there is no solution.

2.There is a time when there is a single solution:

a 2 is not equal to 3 1

i.e. a is not equal to 6

3.Countless solutions to the situation:

a/2=3/1=9/b

That is, when a=6 and b=3, there are an infinite number of solutions.

ax+by=c，①

ax+by=d，②

If c = d, then the binary system of linear equations has an infinite number of solutions;

If c ≠ d, then there is no solution to the binary system of linear equations.