Let's express the equation and substitute instead. Solving systems of equations by substitution method
2. Method of algebraic addition.
3. The method of introducing a new variable (the method of changing a variable).
Definition: A system of equations refers to several equations in one or more variables that must be performed simultaneously, i.e. with the same values of variables for all equations. The equations in the system are combined with the system sign - a curly bracket.
Example 1:
is a system of two equations with two variables x And y.
The solution of the system is the roots. When these values are substituted, the equations turn into true identities:
Solving systems of linear equations.
The most common method for solving a system is the substitution method.
Substitution method.
The substitution method for solving systems of equations consists in expressing some variable from one equation of the system in terms of others, and substituting this expression into the remaining equations of the system instead of the expressed variable.
Example 2:
Solve the system of equations:
Solution:
A system of equations is given and it needs to be solved by the substitution method.
Let's express the variable y from the second equation of the system.
Comment:"Express a variable" means to transform the equality so that this variable remains to the left of the equal sign with a coefficient of 1, and all other terms go to the right side of the equality.
The second equation of the system:
Let's just leave it on the left y:
And let's substitute (that's where the name of the method comes from) into the first equation instead of at the expression it is equal to, i.e. .
First equation:
Substitute : 
Let's solve this banal quadratic equation. For those who have forgotten how to do this, there is an article Solving quadratic equations. . 
So the values of the variable x found.
Substitute these values into the expression for the variable y. There are two values here x, i.e. for each of them it is necessary to find the value y .
1) Let
Substitute in the expression.
2) Let
Substitute in the expression.
Everything can be answered:
Comment: In this case, the answer should be written in pairs, so as not to confuse which value of the variable y corresponds to which value of the variable x.
Answer:
Comment: In example 1, only one pair is indicated as a solution to the system, i.e. this pair is a solution to the system, but not a complete one. Therefore, how to solve an equation or system means to indicate the solution and show that there are no other solutions. And here is another couple.
Let's formalize the solution of this system in a school way:
Comment: The sign "" means "equivalent", i.e. the following system or expression is equivalent to the previous one.
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Place of the lesson in the system of lessons: the third lesson of studying the topic “Systems of two linear equations with two variables"
Lesson type: learning new knowledge
Educational Technology: development of critical thinking through reading and writing
Teaching method: study
Lesson Objectives: master another way to solve systems of linear equations with two variables - the addition method
Tasks:
- subject: the formation of practical skills in solving systems of linear equations by the substitution method;
- metasubject: develop thinking, conscious perception of educational material;
- personal: education of cognitive activity, culture of communication and instilling interest in the subject.
As a result, the student:
- Knows the definition of a system of linear equations with two variables;
- Knows what it means to solve a system of linear equations in two variables;
- Able to write a system of linear equations with two variables;
- Understands how many solutions a system of linear equations with two variables can have;
- Is able to determine whether the system has solutions, and if so, how many;
- Knows the algorithm for solving systems of linear equations by substitution, algebraic addition, graphical method.
Problem question:“How to solve a system of linear equations with two variables?”
Key questions: How and why do we use equations in our lives?
Equipment: presentation; multimedia projector; screen; computer, algebra workbook: grade 7: to the textbook by A.G. Mordkovich and others "Algebra - 7" 2012
Resources (where information on the topic comes from: books, textbooks, Internet, etc.): textbook "Algebra - 7" 2012, A.G. Mordkovich
Forms of organization of educational activities of students (group, pair-group, frontal, etc.): individual, partially frontal, partially steam room
Evaluation criteria:
- A - knowledge and understanding +
- B - application and reasoning
- C - message +
- D - reflection and evaluation
Areas of interaction:
- ATL - Be able to use time effectively, plan your activities in accordance with the goals and objectives set, determine the most rational sequence of activities. Ability to answer questions, argue, argue. To be able to analyze and evaluate their own educational and cognitive activity, to find ways to solve problems.
- HI students explore the consequences of human activities
During the classes
I. Organization of the lesson
II. Self-training check
a) No. 12.2(b, c).
Answer: (5; 3). Answer: (2; 3).
Answer: (4;2)
Express one variable in terms of another:
- p \u003d p / (g * h) - liquid density
- p \u003d g * p * h - liquid pressure at the bottom of the vessel
- h = p / (g * p) - height
- p = m / V - density
- m = V * p -mass
- p = m / V - density
Algorithm for solving a system of two equations with two variables using the substitution method:
- Express y in terms of x from the first (or second) equation of the system.
- Substitute the expression obtained at the first step instead of y into the second (first) equation of the system.
- Solve the equation obtained in the second step for x.
- Substitute the value of x found at the third step into the expression y through x obtained at the first step.
- Write the answer as a pair of values (x; y) that were found in the third and fourth steps, respectively.
Independent work:
In the workbook, pp. 46 - 47.
- on “3” No. 6(a);
- on “4” No. 6(b);
- to "5" No. 7.
III. Updating of basic knowledge
What is a system of linear equations with two variables?
A system of equations is two or more equations for which it is necessary to find all their common solutions.
What is the solution of a system of equations with two variables?
A solution to a system of two equations with two unknowns is a pair of numbers (x, y) such that if these numbers are substituted into the equations of the system, then each of the equations of the system turns into a true equality.
How many solutions can a system of linear equations with two variables have?
If the slopes are equal, then the lines are parallel, there are no roots.
If the slopes are not equal, then the lines intersect, one root (the coordinates of the intersection point).
If the slopes are equal, then the lines coincide, the root is infinite.
IV. Learning new material
Fill in the blanks: Appendix 1 (followed by slide self-examination)
V. Work on the topic of the lesson
In class: Nos. 13.2(a, d), 13.3(a, d).
VI. Homework
Paragraph 13 - textbook; dictionary; No. 13.2(b, c), 13.3(b, c).
VII. Lesson summary
- Hooray!!! I understand everything!
- There are things I need to work on!
- There were failures, but I will overcome everything!
VIII. Solving problems for the military component
Main battle tank T-80.
Adopted in 1976. The world's first serial tank with a main power plant based on a gas turbine engine.
Basic tactical and technical data (TTD):
Weight, t - 46
Speed, km/h - 70
Power reserve, km - 335-370
Armament: 125-mm smooth-bore gun (40 pieces of ammunition);
12.7 mm machine gun (ammunition load 300 pieces);
7.62 mm PKT machine gun (ammunition load 2000 pcs.)
How long can a T-80 tank be in motion without refueling?
In this case, it is convenient to express x through y from the second equation of the system and substitute the resulting expression instead of x into the first equation:
The first equation is an equation with one variable y. Let's solve it:
5(7-3y)-2y = -16
The resulting value of y is substituted into the expression for x:
Answer: (-2; 3).
In this system, it is easier to express y in terms of x from the first equation and substitute the resulting expression instead of y in the second equation:
The second equation is an equation with one variable x. Let's solve it:
3x-4(-1.5-3.5x)=23
In the expression for y, instead of x, we substitute x=1 and find y:
Answer: (1; -5).
Here it is more convenient to express y in terms of x from the second equation (since dividing by 10 is easier than dividing by 4, -9 or 3):
We solve the first equation:
4x-9(1.6-0.3x)= -1
4x-14.4+2.7x=-1
Substitute x=2 and find y:
Answer: (2; 1).
Before applying the substitution method, this system should be simplified. Both parts of the first equation can be multiplied by the least common denominator, in the second equation we open the brackets and give like terms:
We have obtained a system of linear equations with two variables. Now let's apply the substitution. It is convenient to express a in terms of b from the second equation:
We solve the first equation of the system:
3(21.5 + 2.5b) - 7b = 63
It remains to find the value of a:
According to the formatting rules, we write the answer in parentheses separated by a semicolon in alphabetical order.
Answer: (14; -3).
When expressing one variable in terms of another, it is sometimes more convenient to leave it with some coefficient.
Systems of equations are widely used in the economic industry in the mathematical modeling of various processes. For example, when solving problems of production management and planning, logistics routes (transport problem) or equipment placement.
Equation systems are used not only in the field of mathematics, but also in physics, chemistry and biology, when solving problems of finding the population size.
A system of linear equations is a term for two or more equations with several variables for which it is necessary to find a common solution. Such a sequence of numbers for which all equations become true equalities or prove that the sequence does not exist.
Linear Equation
Equations of the form ax+by=c are called linear. The designations x, y are the unknowns, the value of which must be found, b, a are the coefficients of the variables, c is the free term of the equation.
Solving the equation by plotting its graph will look like a straight line, all points of which are the solution of the polynomial.
Types of systems of linear equations
The simplest are examples of systems of linear equations with two variables X and Y.
F1(x, y) = 0 and F2(x, y) = 0, where F1,2 are functions and (x, y) are function variables.
Solve a system of equations - it means to find such values (x, y) for which the system becomes a true equality, or to establish that there are no suitable values of x and y.
A pair of values (x, y), written as point coordinates, is called a solution to a system of linear equations.
If the systems have one common solution or there is no solution, they are called equivalent.
Homogeneous systems of linear equations are systems whose right side is equal to zero. If the right part after the "equal" sign has a value or is expressed by a function, such a system is not homogeneous.
The number of variables can be much more than two, then we should talk about an example of a system of linear equations with three variables or more.
Faced with systems, schoolchildren assume that the number of equations must necessarily coincide with the number of unknowns, but this is not so. The number of equations in the system does not depend on the variables, there can be an arbitrarily large number of them.
Simple and complex methods for solving systems of equations
There is no general analytical way to solve such systems, all methods are based on numerical solutions. IN school course Mathematics describes in detail such methods as permutation, algebraic addition, substitution, as well as the graphical and matrix method, the solution by the Gauss method.
The main task in teaching methods of solving is to teach how to correctly analyze the system and find the optimal solution algorithm for each example. The main thing is not to memorize a system of rules and actions for each method, but to understand the principles of applying a particular method.
Solving examples of systems of linear equations of the 7th class of the program secondary school quite simple and explained in great detail. In any textbook on mathematics, this section is given enough attention. The solution of examples of systems of linear equations by the method of Gauss and Cramer is studied in more detail in the first courses of higher educational institutions.
Solution of systems by the substitution method
The actions of the substitution method are aimed at expressing the value of one variable through the second. The expression is substituted into the remaining equation, then it is reduced to a single variable form. The action is repeated depending on the number of unknowns in the system
Let's give an example of a system of linear equations of the 7th class by the substitution method:
As can be seen from the example, the variable x was expressed through F(X) = 7 + Y. The resulting expression, substituted into the 2nd equation of the system in place of X, helped to obtain one variable Y in the 2nd equation. The solution of this example does not cause difficulties and allows you to get the Y value. The last step is to check the obtained values.
It is not always possible to solve an example of a system of linear equations by substitution. The equations can be complex and the expression of the variable in terms of the second unknown will be too cumbersome for further calculations. When there are more than 3 unknowns in the system, the substitution solution is also impractical.
Solution of an example of a system of linear inhomogeneous equations:
Solution using algebraic addition
When searching for a solution to systems by the addition method, term-by-term addition and multiplication of equations by various numbers are performed. The ultimate goal of mathematical operations is an equation with one variable.
Applications of this method require practice and observation. It is not easy to solve a system of linear equations using the addition method with the number of variables 3 or more. Algebraic addition is useful when the equations contain fractions and decimal numbers.
Solution action algorithm:
- Multiply both sides of the equation by some number. As a result of the arithmetic operation, one of the coefficients of the variable must become equal to 1.
- Add the resulting expression term by term and find one of the unknowns.
- Substitute the resulting value into the 2nd equation of the system to find the remaining variable.
Solution method by introducing a new variable
A new variable can be introduced if the system needs to find a solution for no more than two equations, the number of unknowns should also be no more than two.
The method is used to simplify one of the equations by introducing a new variable. The new equation is solved with respect to the entered unknown, and the resulting value is used to determine the original variable.
It can be seen from the example that by introducing a new variable t, it was possible to reduce the 1st equation of the system to a standard square trinomial. You can solve a polynomial by finding the discriminant.
It is necessary to find the value of the discriminant using the well-known formula: D = b2 - 4*a*c, where D is the desired discriminant, b, a, c are the multipliers of the polynomial. In the given example, a=1, b=16, c=39, hence D=100. If the discriminant is greater than zero, then there are two solutions: t = -b±√D / 2*a, if the discriminant is less than zero, then there is only one solution: x= -b / 2*a.
The solution for the resulting systems is found by the addition method.
A visual method for solving systems
Suitable for systems with 3 equations. The method consists in plotting graphs of each equation included in the system on the coordinate axis. The coordinates of the points of intersection of the curves and will be common solution systems.
The graphic method has a number of nuances. Consider several examples of solving systems of linear equations in a visual way.
As can be seen from the example, two points were constructed for each line, the values of the variable x were chosen arbitrarily: 0 and 3. Based on the values of x, the values for y were found: 3 and 0. Points with coordinates (0, 3) and (3, 0) were marked on the graph and connected by a line.
The steps must be repeated for the second equation. The point of intersection of the lines is the solution of the system.
In the following example, it is required to find a graphical solution to the system of linear equations: 0.5x-y+2=0 and 0.5x-y-1=0.
As can be seen from the example, the system has no solution, because the graphs are parallel and do not intersect along their entire length.
The systems from Examples 2 and 3 are similar, but when constructed, it becomes obvious that their solutions are different. It should be remembered that it is not always possible to say whether the system has a solution or not, it is always necessary to build a graph.
Matrix and its varieties
Matrices are used to briefly write down a system of linear equations. A matrix is a special type of table filled with numbers. n*m has n - rows and m - columns.
A matrix is square when the number of columns and rows is equal. A matrix-vector is a single-column matrix with an infinitely possible number of rows. A matrix with units along one of the diagonals and other zero elements is called identity.
An inverse matrix is such a matrix, when multiplied by which the original one turns into a unit one, such a matrix exists only for the original square one.
Rules for transforming a system of equations into a matrix
With regard to systems of equations, the coefficients and free members of the equations are written as numbers of the matrix, one equation is one row of the matrix.
A matrix row is called non-zero if at least one element of the row is not equal to zero. Therefore, if in any of the equations the number of variables differs, then it is necessary to enter zero in place of the missing unknown.
The columns of the matrix must strictly correspond to the variables. This means that the coefficients of the variable x can only be written in one column, for example the first, the coefficient of the unknown y - only in the second.
When multiplying a matrix, all matrix elements are sequentially multiplied by a number.
Options for finding the inverse matrix
The formula for finding the inverse matrix is quite simple: K -1 = 1 / |K|, where K -1 is the inverse matrix and |K| - matrix determinant. |K| must not be equal to zero, then the system has a solution.
The determinant is easily calculated for a two-by-two matrix, it is only necessary to multiply the elements diagonally by each other. For the "three by three" option, there is a formula |K|=a 1 b 2 c 3 + a 1 b 3 c 2 + a 3 b 1 c 2 + a 2 b 3 c 1 + a 2 b 1 c 3 + a 3 b 2 c 1 . You can use the formula, or you can remember that you need to take one element from each row and each column so that the column and row numbers of the elements do not repeat in the product.
Solution of examples of systems of linear equations by the matrix method
The matrix method of finding a solution makes it possible to reduce cumbersome notations when solving systems with big amount variables and equations.
In the example, a nm are the coefficients of the equations, the matrix is a vector x n are the variables, and b n are the free terms.
Solution of systems by the Gauss method
In higher mathematics, the Gauss method is studied together with the Cramer method, and the process of finding a solution to systems is called the Gauss-Cramer method of solving. These methods are used to find the variables of systems with a large number of linear equations.
The Gaussian method is very similar to substitution and algebraic addition solutions, but is more systematic. In the school course, the Gaussian solution is used for systems of 3 and 4 equations. The purpose of the method is to bring the system to the form of an inverted trapezoid. By algebraic transformations and substitutions, the value of one variable is found in one of the equations of the system. The second equation is an expression with 2 unknowns, and 3 and 4 - with 3 and 4 variables, respectively.
After bringing the system to the described form, the further solution is reduced to the sequential substitution of known variables into the equations of the system.
In school textbooks for grade 7, an example of a Gaussian solution is described as follows:
As can be seen from the example, at step (3) two equations were obtained 3x 3 -2x 4 =11 and 3x 3 +2x 4 =7. The solution of any of the equations will allow you to find out one of the variables x n.
Theorem 5, which is mentioned in the text, states that if one of the equations of the system is replaced by an equivalent one, then the resulting system will also be equivalent to the original one.
The Gaussian method is difficult for middle school students to understand, but is one of the most interesting ways to develop the ingenuity of children studying in the advanced study program in math and physics classes.
For ease of recording calculations, it is customary to do the following:
Equation coefficients and free terms are written in the form of a matrix, where each row of the matrix corresponds to one of the equations of the system. separates the left side of the equation from the right side. Roman numerals denote the numbers of equations in the system.
First, they write down the matrix with which to work, then all the actions carried out with one of the rows. The resulting matrix is written after the "arrow" sign and continue to perform the necessary algebraic operations until the result is achieved.
As a result, a matrix should be obtained in which one of the diagonals is 1, and all other coefficients are equal to zero, that is, the matrix is reduced to a single form. We must not forget to make calculations with the numbers of both sides of the equation.
This notation is less cumbersome and allows you not to be distracted by listing numerous unknowns.
The free application of any method of solution will require care and a certain amount of experience. Not all methods are applied. Some ways of finding solutions are more preferable in a particular area of human activity, while others exist for the purpose of learning.
The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Equations have been used by man since ancient times and since then their use has only increased. The substitution method makes it easy to solve systems of linear equations of any complexity. The essence of the method is that, using the first expression of the system, we express "y", and then we substitute the resulting expression into the second equation of the system instead of "y". Since the equation already contains not two unknowns, but only one, we can easily find the value of this variable, and then use it to determine the value of the second.
Suppose we are given a system of linear equations of the following form:
\[\left\(\begin(matrix) 3x-y-10=0\\ x+4y-12=0 \end(matrix)\right.\]
Express \
\[\left\(\begin(matrix) 3x-10=y\\ x+4y-12=0 \end(matrix)\right.\]
Substitute the resulting expression into the 2nd equation:
\[\left\(\begin(matrix) y=3x-10\\ x+4(3x-10)-12=0 \end(matrix)\right.\]
Find the value \
Simplify and solve the equation by opening the brackets and taking into account the rules for transferring terms:
Now we know the value of \ Let's use this to find the value of \
Answer: \[(4;2).\]
Where can I solve a system of equations online using the substitution method?
You can solve the system of equations on our website. Free online solver will allow you to solve an online equation of any complexity in seconds. All you have to do is just enter your data into the solver. You can also learn how to solve the equation on our website. And if you have any questions, you can ask them in our Vkontakte group.