The coordinates of the point of intersection of two straight lines are examples of finding.

When solving some geometric problems using the coordinate method, one has to find the coordinates of the point of intersection of the lines. Most often it is necessary to look for the coordinates of the point of intersection of two lines on a plane, but sometimes it is necessary to determine the coordinates of the point of intersection of two lines in space. In this article we will just deal with finding the coordinates of the point at which two straight lines intersect.

The point of intersection of two lines is the definition.

Let's first give a definition of the intersection point of two straight lines.

In the section, the mutual arrangement of straight lines on a plane shows that two straight lines on a plane can either coincide (they have infinitely many common points), or be parallel (and the two straight lines do not have common points), or intersect, having one common point. The variants of the mutual arrangement of two straight lines in space are larger - they can coincide (have infinitely many common points), can be parallel (that is, lie in one plane and not intersect), can be crossed (not lying in one plane), and also have one common point, that is, intersect. So, two straight lines both on the plane and in space are called intersecting if they have one common point.

From the definition of intersecting straight lines follows the definition of the intersection point of straight lines : the point at which two straight lines intersect is called the intersection point of these straight lines. In other words, the only common point of two intersecting lines is the intersection point of these lines.

For clarity, we present a graphic illustration of the point of intersection of two straight lines on a plane and in space.

Finding the coordinates of the point of intersection of two straight lines on the plane.

Before finding the coordinates of the intersection point of two lines on the plane by their known equations, consider the auxiliary problem.

Let a rectangular Cartesian coordinate system Oxy be fixed on a plane and two intersecting lines a and b be given . We assume that the line a corresponds to the general equation of the line , and direct b - type . Let be - some point of the plane, and you want to find out whether the point M _{0 is} the intersection point of the given straight lines.

Solve the problem.

If M ₀ is the intersection point of lines a and b , then by definition it belongs to both a and a and b , that is, its coordinates must simultaneously satisfy the equation and equation . Consequently, we need to substitute the coordinates of the point M ₀ into the equations of the given straight lines and see if this results in two true equalities. If the coordinates of the point M ₀ satisfy both equations and then - the point of intersection of the lines a and b , otherwise M _{0 is} not the point of intersection of the lines.

Now we can proceed to the problem of finding the coordinates of the point of intersection of two straight lines by given equations of straight lines on a plane.

Let a rectangular Cartesian coordinate system Oxy be fixed on the plane and two intersecting straight lines a and b are given by the equations and respectively. Denote the intersection point of the given straight lines as M ₀ and solve the following problem: find the coordinates of the intersection point of two straight lines a and b by the known equations of these straight lines and .

The point M ₀ belongs to each of the intersecting lines a and b by definition. Then the coordinates of the intersection point of the lines a and b simultaneously satisfy the equation and equation . Consequently, the coordinates of the intersection point of two straight lines a and b are the solution of the system of equations (See the article Solving Systems of Linear Algebraic Equations).

Thus, to find the coordinates of the intersection point of two lines defined on the plane by general equations, it is necessary to solve a system composed of the equations of the given lines.

Consider the solution of the example.

So, finding the coordinates of the point of intersection of two lines defined by general equations on a plane reduces to solving a system of two linear equations with two unknown variables. But what if the lines on a plane are not given by general equations, but by equations of a different type (see the types of the equation of a line on a plane)? In these cases, you can first reduce the equations of the straight lines to a general form, and after that find the coordinates of the intersection point.

There is another way to find the coordinates of the point of intersection of two straight lines on a plane. It is convenient to use it when one of the lines is given by parametric equations of the form and the other is a direct equation of another type. In this case, in another equation, instead of variables x and y, you can substitute the expression and where you can get the value which corresponds to the intersection point of the given straight lines. The point of intersection of the lines has coordinates .

Find the coordinates of the point of intersection of lines from the previous example in this way.

To complete the picture should discuss one more thing.

Before finding the coordinates of the intersection point of two straight lines on the plane, it is useful to make sure that the given straight lines really intersect. If it turns out that the original lines coincide or are parallel, then finding the coordinates of the intersection point of such lines is out of the question.

You can, of course, do without such a test, and immediately form a system of equations of the form and solve it. If the system of equations has a unique solution, then it gives the coordinates of the point at which the original lines intersect. If the system of equations does not have a solution, then we can conclude that the original straight lines are parallel (since there is no such pair of real numbers x and y that would satisfy both equations of the given straight lines). From the presence of an infinite set of solutions to the system of equations, it follows that the original lines have infinitely many common points, that is, they coincide.

Consider examples suitable for these situations.

Finding the coordinates of the intersection point of two straight lines in space.

The coordinates of the intersection point of two straight lines in three-dimensional space are found similarly.

Let the intersecting straight lines a and b are given in a rectangular coordinate system Oxyz by the equations of two intersecting planes, that is, the straight line a is defined by a system of the form , and direct b - . Let M ₀ be the intersection point of lines a and b . Then the point M _0, by definition, belongs to both the straight line a and the straight line b , therefore, its coordinates satisfy the equations of both straight lines. Thus, the coordinates of the point of intersection of the lines a and b represent a solution of a system of linear equations of the form . Here we find useful information from the section on the resolution of systems of linear equations in which the number of equations does not coincide with the number of unknown variables.

Consider solving examples.

It should be noted that the system of equations has a unique solution if and only if the lines a and b intersect. If the lines a and b are parallel or intersecting, then the latter system of equations has no solution, since in this case the lines have no common points. If the lines a and b coincide, then they have an infinite set of common points, therefore, this system of equations has an infinite set of solutions. However, in these cases we cannot talk about finding the coordinates of the point of intersection of the lines, since the lines are not intersecting.

Thus, if we do not know in advance whether the given straight lines a and b intersect or not, then it is reasonable to make a system of equations of the form and solve it by the Gauss method. If we obtain a unique solution, then it will correspond to the coordinates of the intersection point of lines a and b . If the system proves to be inconsistent, then the lines a and b do not intersect. If the system has an infinite set of solutions, then the lines a and b coincide.

You can do without the use of the Gauss method. Alternatively, one can calculate the ranks of the basic and extended matrices of this system, and based on the data obtained and the Kronecker-Capelli theorem, one can conclude either about the existence of a single solution, or about the existence of a set of solutions, or about the absence of solutions. It's a matter of taste.

When intersecting lines are given by canonical equations of a line in space or by parametric equations of a line in space, you should first obtain their equations in the form of two intersecting planes, and then find the coordinates of the intersection point.