Consider the following equations:
1. 2*x + 3*y = 15;
2. x2 + y2 = 4;
4. 5*x 3 + y 2 = 8.
Each of the above equations is an equation with two variables. Many points coordinate plane, whose coordinates turn the equation into the correct one numerical equality, is called graph of an equation in two unknowns.
Graph of an equation with two variables
Equations with two variables have a wide variety of plots. For example, for the equation 2*x + 3*y = 15, the graph will be a straight line, for the equation x 2 + y 2 = 4, the graph will be a circle with a radius of 2, the graph of the equation y*x = 1 will be a hyperbola, etc.
Integer equations with two variables also have such a thing as a degree. This degree is determined in the same way as for the whole equation with one variable. To do this, the equation is brought to the form when the left side is a polynomial standard view, while the right one is zero. This is done through equivalent transformations.
Graphical way to solve systems of equations
Let's figure out how to solve systems of equations that will consist of two equations with two variables. Consider a graphical way to solve such systems.
Example 1. Solve the system of equations:
( x 2 + y 2 = 25
(y = -x 2 + 2*x + 5.
Let's plot the graphs of the first and second equations in the same coordinate system. The graph of the first equation will be a circle centered at the origin and radius 5. The graph of the second equation will be a parabola with branches down.
All points of the graphs will each satisfy their own equation. We need to find such points that will satisfy both the first and second equations. Obviously, these will be the points where these two graphs intersect.
Using our drawing, we find the approximate values of the coordinates at which these points intersect. We get the following results:
A(-2.2;-4.5), B(0;5), C(2.2;4.5), D(4,-3).
So our system of equations has four solutions.
x1 ≈ -2.2; y1 ≈ -4.5;
x2 ≈ 0; y2 ≈ 5;
x3 ≈ 2.2; y3 ≈ 4.5;
x4 ≈ 4,y4 ≈ -3.
If we substitute these values into the equations of our system, we can see that the first and third solutions are approximate, and the second and fourth are exact. The graphical method is often used to estimate the number of roots and their approximate boundaries. Solutions are more often approximate than exact.
In this lesson, we will consider solving systems of two equations with two variables. First, consider the graphical solution of a system of two linear equations, the specifics of the totality of their graphs. Next, we solve several systems using a graphical method.
Topic: Systems of Equations
Lesson: Graphical method for solving a system of equations
Consider the system
A pair of numbers that is simultaneously a solution to both the first and second equations of the system is called solution of the system of equations.
To solve a system of equations means to find all its solutions, or to establish that there are no solutions. We have considered the graphs of the basic equations, let's move on to the consideration of systems.
Example 1. Solve the system 
Decision:
These are linear equations, the graph of each of them is a straight line. The graph of the first equation passes through the points (0; 1) and (-1; 0). The graph of the second equation passes through the points (0; -1) and (-1; 0). The lines intersect at the point (-1; 0), this is the solution to the system of equations ( Rice. 1).
The solution of the system is a pair of numbers. Substituting this pair of numbers into each equation, we obtain the correct equality.
We got only decision linear system.
Recall that when solving a linear system, the following cases are possible:
the system has a unique solution - the lines intersect,
the system has no solutions - the lines are parallel,
the system has an infinite number of solutions - the lines coincide.
We have reviewed special case systems when p(x; y) and q(x; y) are linear expressions in x and y.
Example 2. Solve a system of equations 
Decision:
The graph of the first equation is a straight line, the graph of the second equation is a circle. Let's build the first graph by points (Fig. 2).
The center of the circle is at the point O(0; 0), the radius is 1.
The graphs intersect at point A(0; 1) and point B(-1; 0).
Example 3. Solve the system graphically
Solution: Let's build a graph of the first equation - this is a circle with a center at point O (0; 0) and a radius of 2. The graph of the second equation is a parabola. It is shifted relative to the origin by 2 upwards, i.e. its top is the point (0; 2) (Fig. 3).
Graphs have one common point- t. A(0; 2). It is the solution to the system. Substitute a couple of numbers into the equation to check the correctness.
Example 4. Solve the system
Solution: Let's build a graph of the first equation - this is a circle with a center at point O (0; 0) and a radius of 1 (Fig. 4).
Let's build a graph of the function This is a broken line (Fig. 5).
Now let's move it down by 1 along the oy axis. This will be the graph of the function
Let's place both graphs in the same coordinate system (Fig. 6).
We get three intersection points - point A (1; 0), point B (-1; 0), point C (0; -1).
We have reviewed graphic method systems solutions. If it is possible to graph each equation and find the coordinates of the intersection points, then this method is quite sufficient.
But often the graphical method makes it possible to find only an approximate solution of the system or answer the question about the number of solutions. Therefore, other methods, more accurate, are needed, and we will deal with them in the next lessons.
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2. Internet project "Tasks" ().
3. Educational portal"I WILL RESOLVE THE USE" ().
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The video lesson "Graphical method for solving systems of equations" presents educational material to explore this topic. Material contains general concept about solving a system of equations, as well as detailed explanation using an example of how the system of equations is solved graphically.
The visual aid uses animation for more convenient and understandable execution of constructions, as well as different ways allocation important concepts and details for an in-depth understanding of the material, its better memorization.
The video tutorial starts by introducing the topic. Pupils are reminded what a system of equations is, and what systems of equations they already had to get acquainted with in the 7th grade. Previously, students had to solve systems of equations of the form ax+by=c. Deepening the concept of solving systems of equations and in order to form the ability to solve them, this video lesson discusses the solution of a system consisting of two equations of the second degree, as well as one equation of the second degree, and the second - of the first degree. Reminds you of what a solution to a system of equations is. The definition of the solution of the system as a pair of values of the variables that invert its equations when substituting into the correct equality is displayed on the screen. In accordance with the definition of the solution of the system, the task is specified. It is displayed on the screen to remember that solving a system means finding suitable solutions or proving their absence.
It is proposed to master the graphical method of solving a certain system of equations. Application this method is considered on the example of solving a system consisting of the equations x 2 + y 2 \u003d 16 and y \u003d - x 2 + 2x + 4. Graphic solution system begins with plotting each of these equations. Obviously, the graph of the equation x 2 + y 2 \u003d 16 will be a circle. The points belonging to this circle are the solution to the equation. Next to the equation, a circle with a radius of 4 is built on the coordinate plane with the center O at the origin. The graph of the second equation is a parabola, the branches of which are lowered down. This parabola is constructed on the coordinate plane, corresponding to the graph of the equation. Any point belonging to the parabola is a solution to the equation y \u003d -x 2 + 2x + 4. It is explained that the solution of a system of equations is points on the graphs that simultaneously belong to the graphs of both equations. This means that the intersection points of the constructed graphs will be solutions to the system of equations.
It is noted that the graphical method consists in finding the approximate value of the coordinates of points located at the intersection of two graphs, which reflect the set of solutions to each equation of the system. The figure marks the coordinates of the found intersection points of two graphs: A, B, C, D[-2;-3.5]. These points are solutions of the system of equations found graphically. You can check their correctness by substituting them into the equation and getting a fair equality. After substituting the points into the equation, it can be seen that some of the points give exact value solutions, and part represents the approximate value of the solution of the equation: x 1 =0, y 1 =4; x 2 \u003d 2, y 2 ≈3.5; x 3 ≈3.5, y 3 \u003d -2; x 4 \u003d -2, y 4 ≈ -3.5.
The video tutorial explains in detail the essence and application of the graphical method for solving a system of equations. This makes it possible to use it as a video aid in an algebra lesson at school when studying this topic. Also, the material will be useful for self-study students and can help explain the topic in distance learning.
First level
Solving equations, inequalities, systems using function graphs. visual guide (2019)
Many tasks that we are used to calculating purely algebraically can be solved much easier and faster, using function graphs will help us with this. You say "how so?" to draw something, and what to draw? Trust me, sometimes it's more convenient and easier. Shall we start? Let's start with equations!
Graphical solution of equations
Graphical solution of linear equations
As you already know, the graph of a linear equation is a straight line, hence the name of this type. Linear equations are quite easy to solve algebraically - we transfer all unknowns to one side of the equation, everything that we know - to the other, and voila! We have found the root. Now I'll show you how to do it graphic way.
So you have an equation:
How to solve it?
Option 1, and the most common is to move the unknowns to one side, and the known to the other, we get:
And now we are building. What did you get?
What do you think is the root of our equation? That's right, the coordinate of the intersection point of the graphs:
Our answer is
That's the whole wisdom of the graphic solution. As you can easily check, the root of our equation is a number!
As I said above, this is the most common option, close to algebraic solution, but it can also be done in a different way. To consider an alternative solution, let's return to our equation:
This time we will not move anything from side to side, but will build graphs directly, as they are now:
Built? Look!
What is the solution this time? All right. The same is the coordinate of the point of intersection of the graphs:
And, again, our answer is .
As you can see, with linear equations everything is extremely simple. It's time to consider something more complicated... For example, graphic solution of quadratic equations.
Graphical solution of quadratic equations
So, now let's start solving the quadratic equation. Let's say you need to find the roots of this equation:
Of course, you can now start counting through the discriminant, or according to the Vieta theorem, but many on the nerves make mistakes when multiplying or squaring, especially if the example is with big numbers, and, as you know, you won’t have a calculator at the exam ... Therefore, let's try to relax a bit and draw while solving this equation.
Graphically find solutions given equation can different ways. Consider various options and you can choose which one you like best.
Method 1. Directly
We just build a parabola according to this equation:
To make it quick, I'll give you one little hint: it is convenient to start the construction by determining the vertex of the parabola. The following formulas will help determine the coordinates of the vertex of the parabola:
You say "Stop! The formula for is very similar to the formula for finding the discriminant "yes, it is, and it is a huge minus"direct" construction of a parabola to find its roots. However, let's count to the end, and then I'll show you how to make it much (much!) easier!
Did you count? What are the coordinates of the vertex of the parabola? Let's figure it out together:
Exactly the same answer? Well done! And now we already know the coordinates of the vertex, and to build a parabola, we need more ... points. What do you think, how many minimum points do we need? Correctly, .
You know that a parabola is symmetrical about its vertex, for example:
Accordingly, we need two more points along the left or right branch of the parabola, and in the future we will symmetrically reflect these points on the opposite side:
We return to our parabola. For our case, the point. We need two more points, respectively, can we take positive ones, but can we take negative ones? What are the best points for you? It is more convenient for me to work with positive ones, so I will calculate with and.
Now we have three points, and we can easily build our parabola by reflecting the last two points about its top:
What do you think is the solution to the equation? That's right, the points at which, that is, and. Because.
And if we say that, then it means that it must also be equal, or.
Just? We have finished solving the equation with you in a complex graphical way, or there will be more!
Of course, you can check our answer algebraically - you can calculate the roots through the Vieta theorem or the Discriminant. What did you get? The same? You see! Now let's see a very simple graphical solution, I'm sure you'll like it very much!
Method 2. Split into several functions
Let's take everything, too, our equation: , but we write it in a slightly different way, namely:
Can we write it like this? We can, since the transformation is equivalent. Let's look further.
Let's build two functions separately:
- - the graph is a simple parabola, which you can easily build even without defining the vertex using formulas and making a table to determine other points.
- - the graph is a straight line, which you can just as easily build by estimating the values and in your head without even resorting to a calculator.
Built? Compare with what I got:
Do you think that in this case are the roots of the equation? Correctly! Coordinates by, which are obtained by crossing two graphs and, that is:
Accordingly, the solution to this equation is:
What do you say? Agree, this solution method is much easier than the previous one and even easier than looking for roots through the discriminant! If so, try this method to solve the following equation:
What did you get? Let's compare our charts:
The graphs show that the answers are:
Did you manage? Well done! Now let's look at the equations a little bit more complicated, namely, the solution of mixed equations, that is, equations containing functions of different types.
Graphical solution of mixed equations
Now let's try to solve the following:
Of course, everything can be brought to common denominator, find the roots of the resulting equation, not forgetting to take into account the ODZ, but again, we will try to solve graphically, as we did in all previous cases.
This time let's plot the following 2 graphs:
- - the graph is a hyperbola
- - a graph is a straight line that you can easily build by estimating the values and in your head without even resorting to a calculator.
Realized? Now start building.
Here's what happened to me:
Looking at this picture, what are the roots of our equation?
That's right, and. Here is the confirmation:
Try plugging our roots into the equation. Happened?
All right! Agree, graphically solving such equations is a pleasure!
Try to solve the equation yourself graphically:
I give you a hint: move part of the equation to right side so that both sides have the simplest functions to build. Got the hint? Take action!
Now let's see what you got:
Respectively:
- - cubic parabola.
- - an ordinary straight line.
Well, we are building:
As you wrote down for a long time, the root of this equation is -.
Having solved this a large number of examples, I'm sure you realized how you can easily and quickly solve equations graphically. It's time to figure out how to decide in a similar way systems.
Graphic solution of systems
The graphical solution of systems is essentially no different from the graphical solution of equations. We will also build two graphs, and their intersection points will be the roots of this system. One graph is one equation, the second graph is another equation. Everything is extremely simple!
Let's start with the simplest - solving systems of linear equations.
Solving systems of linear equations
Let's say we have the following system:
To begin with, we will transform it in such a way that on the left there is everything that is connected with, and on the right - what is connected with. In other words, we write these equations as a function in the usual form for us:
And now we just build two straight lines. What is the solution in our case? Correctly! The point of their intersection! And here you need to be very, very careful! Think why? I'll give you a hint: we're dealing with a system: the system has both, and... Got the hint?
All right! When solving the system, we must look at both coordinates, and not only, as when solving equations! Another important point- write them down correctly and not confuse where we have the value, and where the value is! Recorded? Now let's compare everything in order:
And answers: i. Make a check - substitute the found roots into the system and make sure that we solved it correctly in a graphical way?
Solving systems of nonlinear equations
But what if instead of one straight line, we will have quadratic equation? It's okay! You just build a parabola instead of a straight line! Do not believe? Try to solve the following system:
What is our next step? That's right, write it down so that it is convenient for us to build graphs:
And now it’s all about the small thing - I built it quickly and here’s the solution for you! Building:
Are the graphics the same? Now mark the solutions of the system in the picture and correctly write down the revealed answers!
I've done everything? Compare with my notes:
All right? Well done! You already click on such tasks like nuts! And if so, let's give you a more complicated system:
What are we doing? Correctly! We write the system so that it is convenient to build:
I’ll give you a little hint, since the system looks very complicated! When building graphs, build them "more", and most importantly, do not be surprised at the number of intersection points.
So let's go! Exhaled? Now start building!
Well, how? Beautiful? How many intersection points did you get? I have three! Let's compare our graphs:
Same way? Now carefully write down all the solutions of our system:
Now look at the system again:
Can you imagine that you solved it in just 15 minutes? Agree, mathematics is still simple, especially when looking at an expression, you are not afraid to make a mistake, but you take it and decide! You're a big lad!
Graphical solution of inequalities
Graphical solution of linear inequalities
After last example you have everything on your shoulder! Now exhale - compared to the previous sections, this one will be very, very easy!
We will start, as usual, with a graphical solution linear inequality. For example, this one:
To begin with, we will carry out the simplest transformations - we will open the brackets full squares and add like terms:
The inequality is not strict, therefore - is not included in the interval, and the solution will be all points that are to the right, since more, more, and so on:
Answer:
That's all! Easily? Let's solve a simple inequality with two variables:
Let's draw a function in the coordinate system.
Do you have such a chart? And now we carefully look at what we have in inequality? Smaller? So, we paint over everything that is to the left of our straight line. What if there were more? That's right, then they would paint over everything that is to the right of our straight line. Everything is simple.
All solutions of this inequality are “shaded” orange. That's it, the two-variable inequality is solved. This means that the coordinates and any point from the shaded area are the solutions.
Graphical solution of quadratic inequalities
Now we will deal with how to graphically solve quadratic inequalities.
But before we get straight to the point, let's recap some stuff about the square function.
What is the discriminant responsible for? That's right, for the position of the graph relative to the axis (if you don't remember this, then read the theory about quadratic functions for sure).
In any case, here's a little reminder for you:
Now that we have refreshed all the material in our memory, let's get down to business - we will graphically solve the inequality.
I will tell you right away that there are two options for solving it.
Option 1
We write our parabola as a function:
Using the formulas, we determine the coordinates of the vertex of the parabola (in the same way as when solving quadratic equations):
Did you count? What did you get?
Now let's take two more various points and calculate for them:
We begin to build one branch of the parabola:
We symmetrically reflect our points on another branch of the parabola:
Now back to our inequality.
We need to be less than zero, respectively:
Since in our inequality there is a sign strictly less, we exclude the end points - we “poke out”.
Answer:
Long way, right? Now I will show you a simpler version of the graphical solution using the same inequality as an example:
Option 2
We return to our inequality and mark the intervals we need:
Agree, it's much faster.
Let's write down the answer now:
Consider another solution that simplifies and algebraic part, but the main thing is not to get confused.
Multiply the left and right sides by:
Try to solve the following quadratic inequality on your own in any way you like: .
Did you manage?
See how my chart turned out:
Answer: .
Graphical solution of mixed inequalities
Now let's move on to more complex inequalities!
How do you like this:
Horrible, right? Honestly, I have no idea how to solve this algebraically ... But, it is not necessary. Graphically, there is nothing complicated in this! The eyes are afraid, but the hands are doing!
The first thing we start with is by building two graphs:
I will not write a table for everyone - I'm sure you can do it perfectly on your own (of course, there are so many examples to solve!).
Painted? Now build two graphs.
Let's compare our drawings?
Do you have the same? Fine! Now let's place the intersection points and determine with a color which graph we should have, in theory, should be larger, that is. Look what happened in the end:
And now we just look at where our selected chart is higher than the chart? Feel free to take a pencil and paint over given area! It will be the solution to our complex inequality!
At what intervals along the axis is we higher than? Right, . This is the answer!
Well, now you can handle any equation, and any system, and even more so any inequality!
BRIEFLY ABOUT THE MAIN
Algorithm for solving equations using function graphs:
- Express through
- Define the function type
- Let's build graphs of the resulting functions
- Find the intersection points of the graphs
- Correctly write down the answer (taking into account the ODZ and inequality signs)
- Check the answer (substitute the roots in the equation or system)
For more information about plotting function graphs, see the topic "".
