Graphical solution of systems of inequalities in two variables. Equations and Inequalities in Two Variables How to Solve a System of Inequalities in Two Variables

The video lesson "Systems of inequalities with two variables" contains visual educational material on this topic. The lesson includes a consideration of the concept of solving a system of inequalities with two variables, examples of solving such systems in a graphical way. The task of this video lesson is to form the ability of students to solve systems of inequalities with two variables in a graphical way, to facilitate the understanding of the process of finding solutions to such systems and memorizing the solution method.

Each description of the solution is accompanied by figures that represent the solution to the problem on the coordinate plane. These figures clearly show the features of plotting and the location of the points corresponding to the solution. All important details and concepts are highlighted with color. Thus, the video lesson is a convenient tool for solving the teacher's problems in the classroom, frees the teacher from submitting a standard block of material for conducting individual work with students.

The video tutorial begins with a presentation of the topic and an example of finding solutions to a system consisting of inequalities x<=y 2 и у<х+3. Примером точки, координаты которой удовлетворяют условиям обеих неравенств, является (1;3). Отмечается, что, так как данная пара значений является решением обоих неравенств, то она является одним из множества решений. А все множество решений будет охватывать пересечение множеств, которые являются решениями каждого из неравенств. Данный вывод выделен в рамку для запоминания и указания на его важность. Далее указывается, что множество решений на координатной плоскости представляет собой множество точек, которые являются общими для множеств, представляющих решения каждого из неравенств.

Understanding of the conclusions made about the solution of the system of inequalities is consolidated by considering examples. The first solution to the system of inequalities x 2 + y 2<=9 и x+y>= 2. Obviously, solutions to the first inequality on the coordinate plane include the circle x 2 + y 2 = 9 and the region inside it. This area in the figure is filled with horizontal hatching. The set of solutions to the inequality x + y> = 2 includes the line x + y = 2 and the half-plane located above. This area is also indicated on the plane by strokes in a different direction. You can now determine the intersection of the two decision sets in the figure. It is enclosed in a segment of a circle x 2 + y 2<=9, который покрыт штриховкой полуплоскости x+y>=2.

Next, we analyze the solution of the system of linear inequalities y> = x-3 and y> = - 2x + 4. In the figure, a coordinate plane is drawn next to the task condition. A straight line is built on it, corresponding to the solutions of the equation y = x-3. The area for solving the inequality y> = x-3 will be the area located above this straight line. It is shaded. The set of solutions to the second inequality is located above the line y = -2x + 4. This line is also built on the same coordinate plane and the solution region is hatched. The intersection of two sets is an angle constructed by two straight lines, together with its inner region. The solution area of ​​the system of inequalities is filled with double shading.

When considering the third example, the case is described when the graphs of the equations corresponding to the inequalities of the system are parallel straight lines. It is necessary to solve the system of inequalities y<=3x+1 и y>= 3x-2. A straight line is constructed on the coordinate plane corresponding to the equation y = 3x + 1. The range of values ​​corresponding to solutions of the inequality y<=3x+1, лежит ниже данной прямой. Множество решений второго неравенства лежит выше прямой y=3x-2. При построении отмечается, что данные прямые параллельны. Область, являющаяся пересечением двух множеств решений, представляет собой полосу между данными прямыми.

The video lesson "Systems of inequalities with two variables" can be used as a visual aid in a lesson at school or replace the teacher's explanation in independent study of the material. A detailed understandable explanation of the solution of systems of inequalities on the coordinate plane can help to submit material for distance learning.

Festival of research and creative works of students

"Portfolio"

Equations and inequalities in two variables

and their geometric solution.

Fedorovich Julia

10th grade student

MOU SOSH №26

Supervisor:

Kulpina E.V.

mathematic teacher

MOU SOSH №26

Winter, 2007

Introduction.

2. Equations in two variables, their geometric solution and application.

2.1 Systems of equations.

2.2 Examples of solving equations in two variables.

2.3. Examples of solving systems of equations in two variables.

3. Inequalities and their geometric solution.

3.1. Examples of solving inequalities in two variables

4. A graphical method for solving problems with parameters.

5. Conclusion.

6. List of used literature.

1. Introduction

I took a job on this topic because studying the behavior of functions and plotting their graphs is an important branch of mathematics, and fluency in graphing techniques often helps to solve many problems, and sometimes is the only way to solve them. Also, the graphical method for solving equations allows you to determine the number of roots of the equation, the values ​​of the root, find the approximate, and sometimes exact values ​​of the roots.

In engineering and physics, they are often used precisely in the graphical way of defining functions. A seismologist, analyzing a seismogram, finds out when there was an earthquake, where it happened, determines the strength and nature of the tremors. The doctor who examined the patient can judge the cardiogram by the cardiogram: studying the cardiogram helps to correctly diagnose the disease. The radioelectronic engineer chooses the most suitable mode of its operation according to the characteristics of the semiconductor element. It is easy to increase the number of such examples. Moreover, with the development of mathematics, the penetration of the graphic method into the most diverse areas of human life is growing. In particular, the use of functional dependencies and charting is widely used in economics. This means that the importance of studying the section of mathematics under consideration at school, at a university is also growing, and especially the importance of independent work on it.

With the development of computer technology, with its excellent graphics and high speed of operations, working with graphs of functions has become much more interesting, visual and exciting. Having an analytical view of some dependence, you can build a graph quickly, in the desired scale and color, using various software tools.

Equations in two variables and their geometric solution.

Equation of the form f(x; y)=0 is called an equation in two variables.

A solution to an equation in two variables is an ordered pair of numbers (α, β), when substituting (α - instead of x, β - instead of y) in the equation, the expression makes sense f(α; β)=0

For example, for the equation (( X+1)) 2 + at 2 = 0 the ordered pair of numbers (0; 0) is its solution, since the expression ((0 + 1)
) 2 +0 2 makes sense and is equal to zero, but an ordered pair of numbers (-1; 0) is not a solution, since it is not defined
and therefore the expression ((-1 + 1)) 2 +0 2 is meaningless.

To solve an equation means to find the set of all its solutions.

Equations in two variables can:

a) have one solution. For example, the equation x 2 + y 2 = 0 has one solution (0; 0);

b) have multiple solutions. For example, the given equation (‌‌│ X│- 1) 2 +(│at│- 2) 2 has four solutions: (1; 2), (- 1; 2), (1; -2), (- 1; -2);

c) have no solutions. For example the equation X 2 + at 2 + 1 = 0 has no solutions;

d) have infinitely many solutions. For example, an equation such as x-y + 1 = 0 has infinitely many solutions

Sometimes a geometric interpretation of the equation is helpful. f(x; y)= g(x; y) ... On the coordinate plane hoy the set of all solutions is a set of points. In some cases, this set of points is a certain line, and in this case they say that the equation f(x; y)= g(x; y) there is an equation of this line, for example:

fig. 1 fig. 2 fig. 3

fig. 4 fig. 5 fig. 6

2.1 Systems of equations

Let two equations with unknowns be given x and y

F 1 ( x; y) = 0 andF 2 (x; y)=0

We will assume that the first of these equations defines on the plane of variables X and at line Г 1, and the second - line Г 2. To find the intersection points of these lines, it is necessary to find all pairs of numbers (α, β), such that when replacing the unknown in these equations X by the number α and unknown at by the number β, the correct numerical equalities are obtained. If the problem of finding all such pairs of numbers is posed, then they say that it is required to solve a system of equations and write this system using curly brackets in the following form

A solution to a system is a pair of numbers (α, β) that is a solution to both the first and second equations of the given system.

To solve a system means to find the set of all its solutions, or to prove that there are no solutions.

In some cases, the geometric interpretation of each equation of the system, because the solutions of the system correspond to the points of intersection of the lines given by each equation of the system. Often the geometric interpretation allows only guessing about the number of solutions.

For example, let's find out how many solutions the system of equations has

The first of the equations of the system defines a circle with radius R =
with the center (0; 0), and the second is a parabola, the vertex of which is located at the same point. It is now clear that there are two points of intersection of these lines. Therefore, the system has two solutions - these are (1; 1) and (-1; 1)

Examples of solving equations in two variables

Draw all points with coordinates (x; y) for which equality holds.

1. (x-1) (2y-3) = 0

This equation is equivalent to a combination of two equations

Each of the obtained equations defines a straight line on the coordinate plane.

2. (x-y) (x 2 -4) = 0

The solution to this equation is the set of points of the plane, the coordinates, which are satisfied by the set of equations

On the coordinate plane, the solution will look like this

3.
= x 2

Solution: We will use the definition of the absolute value and replace this equation with an equivalent set of two systems

y = x 2 + 2x y = -x 2 + 2x

X 2 + 2x = 0 x v = 1 y v =1

x (x + 2) = 0

X v = -1 y v =1-2=-1

System solutions examples.

Solve the system graphically:

1)

In each equation, we express the variable y in terms of X and build graphs of the corresponding functions:

y =
+1

a) build a graph of the function y =

Function graph y = + 1 obtained from the schedule at= by shifting two units to the right and one unit up:

y = - 0.5x + 2 is a linear function whose graph is a straight line

The solution to this system is the coordinates of the intersection point of the graphs of functions.

Answer (2; 1)

3. Inequalities and their geometric solution.

Inequality with two unknowns can be represented as follows: f(x; y) > 0, where Z = f(x; y) - function of two arguments X and at... If we consider the equation f(x; y) = 0, then you can build its geometric image, i.e. set of points M (x; y), whose coordinates satisfy this equation. In each of the areas, the function f preserves the sign, it remains to choose those in which f(x; y)>0.

Consider the linear inequality ax+ by+ c> 0. If one of the coefficients a or b nonzero, the equation ax+ by+ c=0 defines a straight line that divides the plane into two half-planes. In each of them, the sign of the function z = ax+ by+ c. To determine the sign, you can take any point of the half-plane and calculate the value of the function z at this point.

For instance:

3x - 2y +6>0.

f(x; y) = 3x- 2y +6,

f(-3;0) = -3 <0,

f(0;0) = 6>0.

The solution to the inequality is the set of points of the right half-plane (filled in in Figure 1)

Rice. one

Inequality │y│ + 0.5 ≤
satisfies the set of points of the plane (x; y), shaded in Figure 2. To construct this area, we will use the definition of the absolute value and methods of plotting a function graph using a parallel transfer of the function graph along the OX or OU axis

R
fig. 2

f(x; y) =

f (0;0) = -1,5<0

f(2;2)= 2,1>0

3.1. Examples of solving inequalities with two variables.

Draw many solutions to inequality

a)

y = x 2 -2x

y = | x 2 -2x |

| y | = | x 2 -2x |

f(x; y)=

f (1;0)=-1<0

f(3;0) = -3<0

f(1;2) =1>0

f(-2;-2) = -6<0

f(1;-2)=1>0

The solution to the inequality is the filled area in Figure 3. To construct this area, we used the methods of plotting with the module

Rice. 3

1)
2)
<0

f (2; 0) = 3> 0

f (0; 2) = - 1<0

f (-2; 0) = 1> 0

f (0; -2) = 3> 0

To solve this inequality, we use the definition of the absolute value

3.2. Examples of solving systems of inequalities.

Draw the set of solutions to a system of inequalities on the coordinate plane

a)

b)

4. Graphical method for solving problems with parameters

Tasks with parameters are called tasks in which functions of several variables are actually involved, of which one variable X is chosen as the independent variable, and the remaining ones play the role of parameters. In solving such problems, graphical methods are especially effective. Let's give examples

The figure shows that the straight y = 4 intersects the graph of the function y =
at three points. Hence, the original equation has three solutions for a = 4.

Find all parameter values a for which the equation X 2 -6 | x | + 5 = a has exactly three distinct roots.

Solution: Plot the function y = x 2 -6x + 5 for X≥0 and mirror it about the ordinate. Family of straight lines parallel to the abscissa axis y = a, crosses the graph at three points at a=5

3. Find all values a, for which the inequality
has at least one positive solution.

Set of coordinate plane points, x coordinate and parameter values a which satisfy this inequality are the union of two regions bounded by parabolas. The solution to this problem is a set of points located in the right half-plane at

x + a + x <2

Lesson topic: Inequalities with two variables.

The purpose of the lesson: Teach students how to solve two-variable inequalities.

Lesson Objectives:

1. Introduce the concept of inequality with two variables. Teach students how to solve inequalities. To form the skills of applying the graphical method when solving inequalities, the ability to show the solution on the coordinate plane.

2. To develop students 'thinking, develop students' practical skills.

3. To educate students for hard work, independence, a responsible attitude to business, initiative and independence in decision-making.

Textbook / Literature: Algebra 9, didactic materials.

During the classes:

1. The concept of inequality in two variables and its solutions.

2. Linear inequality in two variables.

Consider the inequalities: 0.5x 2 -2y + l 20 is an inequality with two variables.

Consider the inequality 0.5x 2 -2y + l

For x = 1, y = 2. We obtain the correct inequality 0.5 1 - 2 2 + 1

A pair of numbers (1; 2), in which the value of x is in the first place, and the value of y is in the second, is called the solution of the inequality 0.5x 2 -2y + l

Definition. A solution to an inequality with two variables is a pair of values ​​of these variables that turns this inequality into a true numerical inequality.

If each solution to an inequality with two variables is represented by a point in the coordinate plane, then we get a graph of this inequality. He is some kind of figure. This figure is said to be given or described by an inequality.

Consider linear inequalities in two variables.

Definition. A linear inequality with two variables is an inequality of the form ax + by c, where x and y are variables, and a, b and c are some numbers.

If, in a linear inequality with two variables, the inequality sign is replaced with an equal sign, then a linear equation will be obtained. The graph of the linear equation ax + by = c, in which a or b is not zero, is a straight line. It splits the set of points of the coordinate plane that do not belong to it into two regions, which are open half-planes.

Using examples, consider how the set of solutions to an inequality in two variables on the coordinate plane is depicted.

Example 1. Let us represent on the coordinate plane the set of solutions of the inequality 2y + 3x≤6.

We build a straight line 2y + 3x = 6, y = 3-1.5x

The straight line splits the set of all points of the coordinate plane into points located below it and points located above it. Let's take from each area a control point: A (1; 1), B (1; 3).

The coordinates of point A satisfy this inequality 2y + 3x≤6, 2 1 + 3 1≤6, 5≤6

The coordinates of point B do not satisfy this inequality 2y + 3x≤6, 2 · 3 + 3 · 1≤6.

This inequality is satisfied by the set of points of the region where point A is located. Shade this region. We have depicted a set of solutions to the inequality 2y + 3x≤6.

To display the set of solutions to the inequality on the coordinate plane, proceed as follows:

1. We build a graph of the function y = f (x), which divides the plane into two areas.

2. Choose any of the obtained areas and consider an arbitrary point in it. We check the satisfiability of the original inequality for this point. If, as a result of the check, a correct numerical inequality is obtained, then we conclude that the original inequality is satisfied in the entire region to which the selected point belongs. Thus, the set of solutions to the inequality is the area to which the selected point belongs. If, as a result of the check, an incorrect numerical inequality is obtained, then the set of solutions to the inequality will be the second region to which the selected point does not belong.

3. If the inequality is strict, then the boundaries of the region, that is, the points of the graph of the function y = f (x), are not included in the set of solutions and the boundary is depicted by a dotted line. If the inequality is not strict, then the boundaries of the region, that is, the points of the graph of the function y = f (x), are included in the set of solutions of this inequality, and the boundary in this case is depicted as a solid line.

Conclusion: - by solving the inequality f (x, y) ˃0,)