First level
Solving equations, inequalities, systems using graphs of functions. 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 in this. You say "how so?" to draw something, and what to draw? Trust me, sometimes it's more convenient and easier. Let's get started? Let's start with the 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 species. Linear equations are easy enough to solve algebraically - we transfer all unknowns to one side of the equation, everything that we know - to the other, and voila! We found the root. Now I'll show you how to do it graphically.
So you have an equation:
How to solve it?
Option 1, and the most common is to transfer unknowns in one direction, and known in the other, we get:
Now we are building. What did you do?
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 all the 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 an algebraic solution, but you can solve it in a different way. To consider an alternative solution, let's return to our equation:
This time we will not transfer anything from side to side, but we will build the graphs directly, as they are now:
Have you built it? We look!
What is the solution this time? Everything is correct. The same is the coordinate of the intersection point 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 difficult ... For example, graphical solution of quadratic equations.
Graphical solution of quadratic equations
So, now let's get down to 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 Vieta's theorem, but many people are on their nerves making mistakes when multiplying or squaring, especially if the example is with large numbers, and, as you know, you will not have a calculator on the exam ... So let's try to relax a bit and draw while solving this equation.
You can graphically find solutions to this equation in different ways. Consider various options, and you yourself will choose which one you like best.
Method 1. Directly
We just build a parabola according to this equation:
To do this quickly, I will give you one little tip: it is convenient to start construction by defining the vertex of the parabola. The following formulas will help to determine the coordinates of the vertex of the parabola:
You will say “Stop! The formula for is very similar to the formula for finding the discriminant "yes, it is, and this is a huge disadvantage of" direct "construction of the parabola in order to find its roots. Nevertheless, let's count to the end, and then I'll show you how to make it much (much!) Easier!
Have you counted? 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. How many points do you think we need? Right, .
You know that a parabola is symmetrical about its vertex, for example:
Accordingly, we need two more points on the left or right branch of the parabola, and in the future we will symmetrically reflect these points to the opposite side:
We return to our parabola. For our case, the point. We need two more points, respectively, can we take positive ones, or can we take negative ones? What points are more convenient for you? It is more convenient for me to work with positive ones, so I will calculate at and.
Now we have three points, and we can safely build our parabola, reflecting the last two points relative to its vertex:
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 finished solving the equation with you in a complex graphical way, or else it will be!
Of course, you can check our answer algebraically - count the roots using Vieta's theorem or the Discriminant. What did you do? Same? You see! Now let's see a very simple graphic solution, I'm sure you will like it very much!
Method 2.Divided into several functions
Let's take all our equation too:, but write it down a little differently, namely:
Can we write it down like that? We can, because the transformation is equivalent. We look further.
Let's construct two functions separately:
- - a graph is a simple parabola that you can easily build even without defining a vertex using formulas and compiling a table to determine other points.
- - a graph is a straight line, which you can just as easily plot, having estimated the values and in your head without even resorting to a calculator.
Have you built it? Compare with what came out for me:
What do you think are the roots of the equation in this case? Right! Coordinates by, which were obtained at the intersection of two graphs and, that is:
Accordingly, the solution to this equation is:
What do you say? You must admit that this solution is much easier than the previous one and even easier than looking for roots through the discriminant! If so, try to solve the following equation in this way:
What did you do? Let's compare our graphs:
The graphs show that the answers are:
Did you manage? Well done! Now let's look at the chuuuut equations a little 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, you can bring everything to a common denominator, find the roots of the resulting equation, while not forgetting to take into account the ODV, but again, we will try to solve it graphically, as we did in all previous cases.
This time, let's build the following 2 graphs:
- - the graph is a hyperbola
- - a graph is a straight line that you can easily plot, having estimated 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 figure, tell me what are the roots of our equation?
That's right, and. Here is the confirmation:
Try plugging our roots into the equation. Happened?
That's right! Agree, it is a pleasure to graphically solve such equations!
Try to solve the equation yourself in a graphical way:
Let me give you a hint: move part of the equation to the right side so that the simplest functions to build are on both sides. Got the hint? Take action!
Now let's see what happened:
Respectively:
- is a cubic parabola.
- - an ordinary straight line.
Well, we build:
As you wrote down for a long time, the root of this equation is -.
Having solved such a large number of examples, I'm sure you realized how easy and quick you can solve equations graphically. It's time to figure out how to solve the system in a similar way.
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:
First, let's transform it so that on the left everything that is connected with, and on the right - that is connected with. In other words, we write these equations as a function in our usual form:
Now we just build two straight lines. What is the solution in our case? Right! The point of their intersection! And here you need to be very, very careful! Think why? Let me give you a hint: we are dealing with a system: the system has both and, and ... Understood the hint?
That's right! When solving the system, we must look at both coordinates, and not just, as when solving equations! Another important point is to write them down correctly and not to confuse where we have the meaning and where the meaning! Did you write it down? Now let's compare everything in order:
And the answers are: and. Make a check - substitute the found roots into the system and make sure if we solved it correctly in a graphical way?
Solving systems of nonlinear equations
But what if instead of one straight line, we have a 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's our next step? That's right, write it down so that it is convenient for us to build graphs:
And now, in general, the matter is small - I built it quickly and here's a solution for you! We build:
Are the graphs the same? Now mark the system solutions in the figure and write down the identified answers correctly!
I've done everything? Compare with my posts:
Is that correct? Well done! You are already clicking such tasks like nuts! And if so, we will give you a more complicated system:
What are we doing? Right! We write the system so that it is convenient to build:
I will give you a little hint, as the system looks well, not very simple! When building graphs, build them "more", and most importantly, do not be surprised by the number of intersection points.
So let's go! Exhaled? Now start building!
How is it? Beautiful? How many intersection points did you get? I have three! Let's compare our charts:
Same way? Now, carefully write down all the decisions of our system:
Now take another look at the system:
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 the last example, you can do everything! Exhale now - compared to the previous sections, this one will be very, very light!
We begin, as usual, with a graphical solution to a linear inequality. For example, this one:
To begin with, we will carry out the simplest transformations - we will open the brackets of perfect squares and give similar 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 there is more, more, and so on:
Answer:
That's all! Easily? Let's solve a simple two-variable inequality:
Let's draw a function in the coordinate system.
Have you got such a schedule? And now we are carefully looking at what we have there in inequality? Smaller? So, we paint over everything that is to the left of our straight line. And if there were more? That's right, then they would paint over everything that is to the right of our straight line. It's simple.
All solutions to this inequality are shaded in orange. That's it, the two-variable inequality is solved. This means that the coordinates of any point from the shaded area are the solutions.
Graphical solution of square inequalities
Now we will deal with how to graphically solve square inequalities.
But before we get down to business, let's review some material regarding the square function.
And 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 exactly the theory of quadratic functions).
Anyway, here's a little reminder sign:
Now that we have refreshed all the material in our memory, let's get down to business - we will graphically solve the inequality.
I'll 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):
Have you counted? What did you do?
Now let's take two more different points and calculate for them:
We start building one branch of the parabola:
We symmetrically reflect our points to another branch of the parabola:
Now, back to our inequality.
We need it to be less than zero, respectively:
Since in our inequality the sign is strictly less, then we exclude the end points - "gouge out".
Answer:
Long way, right? Now I will show you a simpler version of a graphical solution using the example of the same inequality:
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:
Let's consider another solution that simplifies the algebraic part, but the main thing is not to get confused.
Let's multiply the left and right sides by:
Try to solve the following square inequality yourself in any way you like:.
Did you manage?
See how the graph turned out for me:
Answer: .
Graphical solution of mixed inequalities
Now let's move on to more complex inequalities!
How do you like this:
Creepy, right? To be honest, I have no idea how to solve this algebraically ... But, it is not necessary. Graphically, there is nothing complicated about it! The eyes are afraid, but the hands are doing!
The first thing we will start with is by plotting two graphs:
I will not paint a table for each one - I am sure you can handle it perfectly on your own (still, there are so many examples to solve!).
Have you painted it? Now build two graphs.
Let's compare our drawings?
Is it the same for you? Fine! Now we will place the intersection points and determine by color which graph we have, in theory, should be larger, that is. See what happened in the end:
And now we just look, where is the selected chart higher than the chart? Feel free to take a pencil and paint over this area! She will be the solution to our complex inequality!
At what intervals along the axis is it 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 graphs of functions:
- Let us express in terms of
- Define the type of the function
- Let's build the 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 details about plotting functions, see the topic "".
Graphical way to solve systems of equations
(9th grade)
Textbook: Algebra, grade 9, edited by S.A. Telyakovsky.
Lesson type: lesson in the complex application of knowledge, skills, and abilities.
Lesson objectives:
Educational: To develop the ability to independently apply knowledge in a complex, to transfer it to new conditions, including working with a computer program for plotting function graphs and finding the number of roots in given equations.
Developing: To form students' ability to highlight the main features, establish similarities and differences. Enrich your vocabulary. Develop speech, complicating its semantic function. To develop logical thinking, cognitive interest, culture of graphic construction, memory, curiosity.
Educational: To foster a sense of responsibility for the result of their work. Teach to empathize with the successes and failures of classmates.
Means of education : computer, multimedia projector, handouts.
Lesson plan:
Organizing time. Homework - 2 min.
Actualization, repetition, correction of knowledge - 8 min.
Learning new material - 10 min.
Practical work - 20 min.
Summing up - 4 min.
Reflection - 1 min.
DURING THE CLASSES
Organizational moment - 2 min.
Hello guys! Today is a lesson on an important topic: "Solving systems of equations."
There are no such areas of knowledge in the exact sciences, wherever this topic is applied. The epigraph to our lesson is the following words : “The mind is not only in knowledge, but also in the ability to apply knowledge in practice ". (Aristotle)
Statement of the topic, goals and objectives of the lesson.
The teacher informs the class about what will be studied in the lesson and sets the task of learning how to solve systems of equations with two variables in a graphical way.
Assignment for home (P.18 No. 416, 418, 419 a).
Repetition of theoretical material - 8 min.
A) Mathematic teacher: Answer questions and justify your answer using ready-made drawings.
1). Find the graph of a quadratic function D = 0 (Students answer the question and name graph 3c).
2). Find the graph of the inverse - proportional function for k> 0 (Students answer the question, call graph 3a ).
3). Find a graph of a circle centered at O (-1; -5). (Students answer the question, they call graph 1b).
4). Find the graph of the function y = 3x -2. (Students answer the question and name graph 3b).
5). Find the graph of a quadratic function D> 0, a> 0. (Students answer the question and call graph 1a ).
Mathematic teacher: – In order to successfully solve systems of equations, let's remember:
1). What is called a system of equations? (A system of equations is called several equations for which it is required to find the values of unknowns that satisfy all these equations simultaneously).
2). What does it mean to solve a system of equations? (To solve a system of equations means to find all the solutions or to prove that there are no solutions).
3). What is called solving a system of equations? (A solution to a system of equations is called a pair of numbers (x; y), at which all the equations of the system turn into true equalities).
4) Find out if the solution to the system of equations 
a pair of numbers: a) x = 1, y = 2;(–)
b) x = 2, y = 4; (+)
c) x = - 2, y = - 4? (+)
III New material - 10 min.
Clause 18 of the textbook is presented by the method of conversation.
Mathematic teacher: In the 7th grade algebra course, we considered systems of equations of the first degree. Now let's deal with the solution of systems composed of equations of the first and second degree.
1. What is called a system of equations?
2. What does it mean to solve a system of equations?
We know that the algebraic method allows you to find exact solutions to the system, and the graphical method allows you to visually see how many roots the system has and find them approximately. Therefore, we will continue to learn to solve systems of equations of the second degree in the next lessons, and today the main goal of the lesson will be the practical use of a computer program for plotting function graphs and finding the number of roots of systems of equations.
IV . Practical work - 20 min. Solving systems of equations graphically. Determination of the roots of equations.(Plotting a graph on a computer.)
The assignments are completed by students on computers. Solutions are checked on the fly.
y = 2x 2 + 5x +3
y = 4
y = -2x 2 + 5x + 3
y = -3x + 4
y = -2x 2 -5x-3
y = -4 + 2x
y = 4x 2 + 5x +3
y = 2
y= -4 x 2 + 5x + 3
y = -3x + 2
y = -4x 2 -5x-3
y = -2 + 2x
y = 4 x 2 + 5 x+5
y = 3
y = -4x 2 + 5x + 5
y = -x + 3
y = -4x 2 -5x-5
y = -2 + 3x
Before you graphs of two equations. Write down the system defined by these equations and its solution.
– Which of the following systems can be solved using this figure?
– 4 systems were given, they had to be correlated with the graphs. Now the task is reversed: yes charts, they need to be correlated with the system.
Summing up the lesson. Grading - 4 min.
* Solving systems of equations. ( Star Assignments *.)
Equations for the 1st group of students:
Equations for the 2nd group of students:
Equations for the 3rd group of students:
x y = 6
x 2 + y = 4
x 2 + y = 3
x - y + 1 = 0
x 2 - y = 3
Date: ________________
Subject: algebra
Topic: "A graphical way to solve systems of equations."
Goals: Use graphs to solve systems of equations.
Tasks:
Educational: teach to solve systems of linear equations with two variables graphically.
Developing: development of students' research abilities, self-control, speech.
Educational: education of a culture of communication, accuracy.
Lesson type: combined
Forms: Frontal survey, work in pairs.
During the classes:
Organizational stage. Communication of the topic of the lesson, setting the goals of the lesson.(write down the number, topic in a notebook)
Homework check (analysis of unsolved problems);
Control of the assimilation of the material:
Repetition and consolidation of the passed material:
| Option number 1 | Option number 2 |
| Plot the function: (xy-1) (x + 1) = 0 (x-2) 2 + (y + 1) 2 = 4 | Plot the function: (xy + 1) (y-1) = 0 (x-1) 2 + (y + 2) 2 = 4 |
Basic knowledge update:
Determination of a linear equation in two variables.
What is called solving a linear equation in two variables?
What is called a graph of a linear equation in two variables?
What is a graph of a linear equation in two variables?
How many points defines a line?
What does it mean to solve a system of equations?
What is called solving a system of linear equations in two variables?
When do two straight lines on a plane intersect?
When are two lines in a plane parallel?
When do two lines on a plane coincide?
Learning new material:
Consider system of two equations with two unknowns. Decision systems of equations are called pair of valuesvariables, who pay each equation of the system into a true equality. Solving a system of equations means finding all its solutions or proving that there are no solutions.
One of the most effective and intuitive ways to solve and study equations and systems of equations graphical way.
Algorithm for plotting an equation with two variables.
Express the variable y in terms of x.
"Take" the points defining the graph.
Plot Equation
Algorithm for solving a system of equations with two variables graphically.
Plot graphs for each of the equations in the system.
Find the coordinates of the intersection point.
Record your answer.
Example 1
Let's solve the system of equations: 
Let us construct in one coordinate system the graphics of the first NS 2
+
y 2 = 25
(circle) and second hu= 12 (hyperbola) equations. It's clear that
the graphs of the equations intersect at four points A(3;
4), V(4;
3)
C (-3; -4) and D (-4;
3), whose coordinates are solutions
one system.
T 
Since the solutions can be found with some accuracy in the graphical method, they must be verified by substitution.
The check shows that the system really has four solutions: (3; 4), (4; 3), (- 3; -4), (- 4; -3).
Lesson assignment:# 415 (b); No. 416; No. 419 (b); No. 420 (b); No. 421 (a, b); No. 422 (a); # 424 (b); No. 426 p. 115-117.
Summarize (estimates).
Reflection.
Let us repeat the algorithm for solving systems of equations in a graphical way.
How many solutions can a system of equations have?
Who learned to solve systems of l equations graphically?
Who hasn't learned?
Who else doubts?
Hands up, who liked the lesson? Who is not? Who is indifferent?
Homework:§18 pp. 114-115 learn the rules.
§17 pp. 108-110 repeat the rules.
ALGEBRA CLASS 9
Graphical way
solving systems of equations
1. Find by the graph:
a) zeros of the function;
b) the range of values of the function;
c) intervals of increase and decrease of the function;
c) intervals in which y ≤0, y≥0.
d ) the smallest value of the function.
1.From the proposed formulas, select the formula
which defines the function represented in the graph
a ) y = - 3x + 1; b) y = 2x + 1;
c) y = 3x + 1 .
From the proposed formulas, select the formula that
sets the function represented in the graph
b) y = - 2x 2 ; c) y = x 2 +1.
a) y = x 2 ;
From the proposed formulas, select the formula that sets the function shown in the graph.
b) y = 2 x 3; c) y = x 3
a) y = 0.5x 3;
From the proposed formulas, select the formula that sets the function shown in the graph
a) y = 4 / x; b) y = - 4 / x;
Linear equation with
one variable
ax = b
- Linear equation with
two variables
Equation in two variables
The graph of an equation with two variables is the set of points of the coordinate plane, the coordinates of which turn the equation into a true equality
The equation
Express y through x
3x + 2y = 6
2y-x 2 =0
This formula sets ... ..
The graph is
2x + y = 0
hyperbola
quadratic
function
y = -1.5x + 3
Linear
function
straight
y = 0.5 x 2
reverse
proportion
y = -2x
parabola
straight, pr-i
through the beginning. coord.
straight
proportion
Ellipse
NS 2 y = 4 (2-y),
y = 8 / (x 2 +4)
System of equations and its solution
Definitions
- A system of equations is a number of equations united by a curly brace. The curly brace means all equations must be executed at the same time
- A solution to a system of equations in two variables is a pair of values of variables that turn each equation in the system into a true equality
- To solve a system of equations means to find all its solutions or to establish that they do not exist.
Way
substitutions
Way
additions
Methods for solving systems of equations
Way
substitutions
Way
additions
Graphical way
solving systems of equations
1. Express y through x in each equation.
2. Build a graph in one coordinate system
each equation.
3. Express y through x in each equation.
4. Build a graph in one coordinate system
each equation
5. Determine the coordinates of the intersection point
charts.
6. Write down the answer: x = ...; y = ..., or (x; y)
System solution graphically
Let us express at
Let's build a graph
first equation
Let's plot the second
equations-circle with
center at point O (0; 0) and
radius 2.
System solution graphically
Let us express at
Let's build a graph
first equation
Let's plot the second
equations-circle with
center at point O (0; 0) and
radius 2.
NS 2 + y 2 =4*
The system has 2 solutions:
Answer: (0; 2), (-2; 0)
1.We start charging,
We knead our hands
Stretch your back, shoulders,
To make it easier for us to sit
2. Twist-twist the head.
Stretch your neck, stop!
One, two, three - tilt to the right,
One, two, three - now to the left.
3. Now stop!
Raise our hands higher
Inhale and exhale. We breathe deeper.
And now we'll sit down at the desks.
Back forward
Attention! Slide previews are for informational purposes only and may not represent all the presentation options. If you are interested in this work, please download the full version.
Goals and objectives of the lesson:
- continue to work on the formation of skills for solving systems of equations by the graphical method;
- conduct research and draw conclusions about the number of solutions to a system of two linear equations;
- develop interest in the subject through play.
DURING THE CLASSES
1. Organizational moment (Plannerka)- 2 minutes.
- Good day! Let's start our traditional planning meeting. We are glad to welcome everyone who is our guest today in our laboratory (I represent the guests). Our laboratory is called: "WORK with interest and pleasure"(showing slide 2). The name serves as a motto in our work. “Create, Decide, Learn, Achieve with interest and pleasure". Dear guests, I present to you the heads of our laboratory (slide 3).
Our laboratory is engaged in the study of scientific papers, research, expertise, works on the creation of creative projects.
Today the topic of our discussion is "Graphical solution of systems of linear equations." (I suggest you write down the topic of the lesson)
Day program:(slide 4)
1. Planner
2. Extended Academic Council:
- Speeches on the topic
- Work permit
3. Expertise
4. Research and discovery
5. Creative project
6. Report
7. Planning
2. Interview and oral work (Extended Academic Council)- 10 min.
- Today we are holding an extended scientific council, which is attended not only by the heads of departments, but also by all members of our team. The laboratory has just begun work on the topic: "Graphical solution of systems of linear equations". We must try to achieve the highest achievements in this matter. Our laboratory should be renowned for the quality of research on this topic. As a Senior Researcher, I wish everyone the best of luck!
The research results will be reported to the head of the laboratory.
The floor for the report on the solution of systems of equations has ... (I call the student to the blackboard). I give the assignment the task (card 1).
And the laboratory assistant ... (I say the last name) will remind you how to build a graph of a function with a module. I give card 2.
Card 1(solution of the task on slide 7)
Solve the system of equations:
Card 2(solving the problem on slide 9)
Plot the function: y = | 1.5x - 3 |
While the staff prepares for the report, I will check how you are ready to do the research. Each of you must be admitted to work. (We start oral counting by writing down answers in a notebook)
Work permit(tasks on slides 5 and 6)
1) Express at across x:
3x + y = 4 (y = 4 - 3x)
5x - y = 2 (y = 5x - 2)
1 / 2y - x = 7 (y = 2x + 14)
2x + 1 / 3y - 1 = 0 (y = - 6x + 3)
2) Solve the equation:
5x + 2 = 0 (x = - 2/5)
4x - 3 = 0 (x = 3/4)
2 - 3x = 0 (x = 2/3)
1 / 3x + 4 = 0 (x = - 12)
3) A system of equations is given:
Which of the pairs of numbers (- 1; 1) or (1; - 1) is the solution to this system of equations?
Answer: (1; - 1)
Immediately after each fragment of oral counting, students exchange notebooks (with a student sitting next to him in the same department), the correct answers appear on the slides; the verifier puts a plus or a minus. At the end of the work, the heads of departments enter the results into a summary table (see below); for each example 1 point is given (it is possible to get 9 points).
Those who scored 5 or more points receive admission to work. The rest receive a conditional tolerance, i.e. will have to work under the supervision of the head of the department.
Table (to be filled in by the boss)
(Tables are given before the start of the lesson)
After obtaining admission, we listen to the students' answers at the blackboard. For the answer, the student receives 9 points if the answer is complete (the maximum number for admission), 4 points if the answer is not complete. Points are entered in the "tolerance" column.
If the solution is correct on the board, then slides 7 and 9 do not need to be shown. If the solution is correct, but not clearly executed, or the solution is incorrect, then the slides must be shown with explanations.
I show slide 8 after the student's answer on card 1. On this slide, conclusions are important for the lesson.
Algorithm for solving systems graphically:
- Express y in terms of x in each equation in the system.
- Plot each equation in the system.
- Find the coordinates of the intersection points of the graphs.
- Make a check (I draw the students' attention to the fact that the graphical method usually gives an approximate solution, but if the intersection of the graphs hits a point with integer coordinates, you can check and get an exact answer).
- Record your answer.
3. Exercises (Expertise)- 5 minutes.
Gross mistakes were made in the work of some employees yesterday. Today you are already more competent in the matter of a graphical solution. You are invited to conduct an examination of the proposed solutions, i.e. find errors in solutions. Slide 10 is shown.
The work is going on in the departments. (Photocopies of assignments with errors are issued on each table; in each department, employees must find errors and highlight them or correct them; hand over the photocopies to the senior researcher, i.e. the teacher). For those who find and correct the mistake, the boss adds 2 points. Then we discuss the mistakes made and indicate them on slide 10.
Error 1
Solve the system of equations:
Answer: There are no solutions.
Students should continue straight to the intersection and receive the answer: (- 2; 1).
Mistake 2.
Solve the system of equations:
Answer: (1; 4).
Students must find the error in the transformation of the first equation and correct it on the finished drawing. Get another answer: (2; 5).
4. Explanation of the new material (Research and discoveries)- 12 minutes
I suggest that students solve three systems graphically. Each student decides independently in a notebook. Only those with conditional admission can be consulted.
Solution
Without plotting graphs, it is clear that the straight lines will coincide.
Slide 11 shows the solution of the systems; it is expected that students will have difficulty writing down the answer in example 3. After working in the departments, we check the solution (for the correct boss adds 2 points). Now it's time to discuss how many solutions a system of two linear equations can have.
Students must draw their own conclusions and explain them by listing the cases of the mutual arrangement of straight lines on the plane (slide 12).
5. Creative project (Exercises)- 12 minutes
The task is given for the department. The chief gives each laboratory assistant, according to his ability, a fragment of its implementation.
Solve systems of equations graphically:
After opening the brackets, students should receive the system:
After expanding the parentheses, the first equation is: y = 2 / 3x + 4.
6. Report (check the execution of the task)- 2 minutes.
After completing the creative project, students turn in their notebooks. On slide 13 I show what should have happened. The chiefs hand over the table. The last column is filled in by the teacher and puts a mark (marks can be reported to students in the next lesson). In the project, the solution to the first system is evaluated with three points, and the second - four.
7. Planning (debriefing and homework)- 2 minutes.
Let's summarize the results of our work. We did a good job. Specifically, we'll talk about the results tomorrow at the planning meeting. Of course, all laboratory assistants, without exception, have mastered the graphical method for solving systems of equations, learned how many solutions a system can have. Tomorrow each of you will have a personal project. For additional preparation: p. 36; 647-649 (2); repeat analytical methods for solving systems. 649 (2) solve also by the analytical method.
Our work was supervised throughout the day by the director of the laboratory, Noman Nou Manovich. His word. (I show the final slide).
Approximate Grading Scale
| Mark | Tolerance | Expertise | Study | Project | Total |
| 3 | 5 | 2 | 2 | 2 | 11 |
| 4 | 7 | 2 | 4 | 3 | 16 |
| 5 | 9 | 3 | 5 | 4 | 21 |