Solve a system of equations using online substitution. Solving systems of equations using the addition method

Algebraic addition method

You can solve a system of equations with two unknowns different ways- graphical method or variable replacement method.

In this lesson we will get acquainted with another method of solving systems that you will probably like - this is the method of algebraic addition.

Where did the idea of putting something in systems come from? When solving systems main problem is the presence of two variables, because we do not know how to solve equations with two variables. This means that one of them must be excluded in some legal way. And such legitimate ways are mathematical rules and properties.

One of these properties is: the sum of opposite numbers is zero. This means that if one of the variables has opposite coefficients, then their sum will be equal to zero and we will be able to exclude this variable from the equation. It is clear that we do not have the right to add only terms with the variable we need. You need to add the entire equations, i.e. separately add similar terms on the left side, then on the right. As a result, we get a new equation containing only one variable. Let's look at what has been said with specific examples.

We see that in the first equation there is a variable y, and in the second there is the opposite number -y. This means that this equation can be solved by addition.

One of the equations is left as it is. Any one you like best.

But the second equation will be obtained by adding these two equations term by term. Those. We add 3x with 2x, we add y with -y, we add 8 with 7.

We obtain a system of equations

The second equation of this system is a simple equation with one variable. From it we find x = 3. Substituting the found value into the first equation, we find y = -1.

Answer: (3; - 1).

Sample design:

Solve a system of equations using the algebraic addition method

There are no variables with opposite coefficients in this system. But we know that both sides of the equation can be multiplied by the same number. Let's multiply the first equation of the system by 2.

Then the first equation will take the form:

Now we see that the variable x has opposite coefficients. This means that we will do the same as in the first example: we will leave one of the equations unchanged. For example, 2y + 2x = 10. And we get the second by addition.

Now we have a system of equations:

We easily find from the second equation y = 1, and then from the first equation x = 4.

Sample design:

Let's summarize:

We learned to solve systems of two linear equations with two unknowns using the algebraic addition method. Thus, we now know three main methods for solving such systems: graphical, variable replacement method and addition method. Almost any system can be solved using these methods. In more difficult cases A combination of these techniques is used.

List of used literature:

Mordkovich A.G., Algebra 7th grade in 2 parts, Part 1, Textbook for general education institutions / A.G. Mordkovich. – 10th ed., revised – Moscow, “Mnemosyne”, 2007.
Mordkovich A.G., Algebra 7th grade in 2 parts, Part 2, Problem book for educational institutions / [A.G. Mordkovich and others]; edited by A.G. Mordkovich - 10th edition, revised - Moscow, “Mnemosyne”, 2007.
HER. Tulchinskaya, Algebra 7th grade. Blitz survey: a manual for students of general education institutions, 4th edition, revised and expanded, Moscow, Mnemosyne, 2008.
Alexandrova L.A., Algebra 7th grade. Thematic testing work V new form for students of general education institutions, edited by A.G. Mordkovich, Moscow, “Mnemosyne”, 2011.
Alexandrova L.A. Algebra 7th grade. Independent work for students of general education institutions, edited by A.G. Mordkovich - 6th edition, stereotypical, Moscow, “Mnemosyne”, 2010.

Let us analyze two types of solutions to systems of equations:

1. Solving the system using the substitution method.
2. Solving the system by term-by-term addition (subtraction) of the system equations.

In order to solve the system of equations by substitution method you need to follow a simple algorithm:
1. Express. From any equation we express one variable.
2. Substitute. We substitute the resulting value into another equation instead of the expressed variable.
3. Solve the resulting equation with one variable. We find a solution to the system.

To solve system by term-by-term addition (subtraction) method need to:
1. Select a variable for which we will make identical coefficients.
2. We add or subtract equations, resulting in an equation with one variable.
3. Solve the resulting linear equation. We find a solution to the system.

The solution to the system is the intersection points of the function graphs.

Let us consider in detail the solution of systems using examples.

Example #1:

Let's solve by substitution method

Solving a system of equations using the substitution method

2x+5y=1 (1 equation)
x-10y=3 (2nd equation)

1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, which means that it is easiest to express the variable x from the second equation.
x=3+10y

2.After we have expressed it, we substitute 3+10y into the first equation instead of the variable x.
2(3+10y)+5y=1

3. Solve the resulting equation with one variable.
2(3+10y)+5y=1 (open the brackets)
6+20y+5y=1
25y=1-6
25y=-5 |: (25)
y=-5:25
y=-0.2

The solution to the equation system is the intersection points of the graphs, therefore we need to find x and y, because the intersection point consists of x and y. Let's find x, in the first point where we expressed it we substitute y.
x=3+10y
x=3+10*(-0.2)=1

It is customary to write points in the first place we write the variable x, and in the second place the variable y.
Answer: (1; -0.2)

Example #2:

Let's solve using the term-by-term addition (subtraction) method.

Solving a system of equations using the addition method

3x-2y=1 (1 equation)
2x-3y=-10 (2nd equation)

1. We choose a variable, let’s say we choose x. In the first equation, the variable x has a coefficient of 3, in the second - 2. We need to make the coefficients the same, for this we have the right to multiply the equations or divide by any number. We multiply the first equation by 2, and the second by 3 and get a total coefficient of 6.

3x-2y=1 |*2
6x-4y=2

2x-3y=-10 |*3
6x-9y=-30

2. Subtract the second from the first equation to get rid of the variable x. Solve the linear equation.
__6x-4y=2

5y=32 | :5
y=6.4

3. Find x. We substitute the found y into any of the equations, let’s say into the first equation.
3x-2y=1
3x-2*6.4=1
3x-12.8=1
3x=1+12.8
3x=13.8 |:3
x=4.6

The intersection point will be x=4.6; y=6.4
Answer: (4.6; 6.4)

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Using the addition method, the equations of a system are added term by term, and one or both (several) equations can be multiplied by any number. As a result, they come to an equivalent SLE, where in one of the equations there is only one variable.

To solve the system method of term-by-term addition (subtraction) follow these steps:

1. Select a variable for which the same coefficients will be made.

2. Now you need to add or subtract the equations and get an equation with one variable.

System solution- these are the intersection points of the function graphs.

Let's look at examples.

Example 1.

Given system:

Having analyzed this system, you can notice that the coefficients of the variable are equal in magnitude and different in sign (-1 and 1). In this case, the equations can be easily added term by term:

We perform the actions circled in red in our minds.

The result of term-by-term addition was the disappearance of the variable y. This is precisely the meaning of the method - to get rid of one of the variables.

-4 - y + 5 = 0 → y = 1,

In system form, the solution looks something like this:

Answer: x = -4 , y = 1.

Example 2.

Given system:

In this example, you can use the “school” method, but it has a rather big disadvantage - when you express any variable from any equation, you will get a solution in ordinary fractions. But solving fractions takes a lot of time and the likelihood of making mistakes increases.

Therefore, it is better to use term-by-term addition (subtraction) of equations. Let's analyze the coefficients of the corresponding variables:

You need to find a number that can be divided by 3 and on 4 , and it is necessary that this number be the minimum possible. This least common multiple. If it's hard for you to choose suitable number, then you can multiply the coefficients: .

Next step:

We multiply the 1st equation by ,

We multiply the 3rd equation by ,

With this video I begin a series of lessons dedicated to systems of equations. Today we will talk about solving systems of linear equations addition method- this is one of the most simple ways, but at the same time one of the most effective.

The addition method consists of three simple steps:

Look at the system and choose a variable that has identical (or opposite) coefficients in each equation;
Perform algebraic subtraction (for opposite numbers - addition) of equations from each other, and then bring similar terms;
Solve the new equation obtained after the second step.

If everything is done correctly, then at the output we will get a single equation with one variable— it won’t be difficult to solve it. Then all that remains is to substitute the found root into the original system and get the final answer.

However, in practice everything is not so simple. There are several reasons for this:

Solving equations using the addition method implies that all lines must contain variables with equal/opposite coefficients. What to do if this requirement is not met?
Not always, after adding/subtracting equations in the indicated way, we get a beautiful construction that can be easily solved. Is it possible to somehow simplify the calculations and speed up the calculations?

To get the answer to these questions, and at the same time understand a few additional subtleties that many students fail at, watch my video lesson:

With this lesson we begin a series of lectures devoted to systems of equations. And we will start from the simplest of them, namely those that contain two equations and two variables. Each of them will be linear.

Systems is 7th grade material, but this lesson will also be useful for high school students who want to brush up on their knowledge of this topic.

In general, there are two methods for solving such systems:

Addition method;
A method of expressing one variable in terms of another.

Today we will deal with the first method - we will use the method of subtraction and addition. But to do this, you need to understand the following fact: once you have two or more equations, you can take any two of them and add them to each other. They are added member by member, i.e. “X’s” are added to “X’s” and similar ones are given, “Y’s” with “Y’s” are similar again, and what is to the right of the equal sign is also added to each other, and similar ones are also given there.

The results of such machinations will be a new equation, which, if it has roots, they will certainly be among the roots of the original equation. Therefore, our task is to do the subtraction or addition in such a way that either $x$ or $y$ disappears.

How to achieve this and what tool to use for this - we’ll talk about this now.

Solving easy problems using addition

So, we learn to use the addition method using the example of two simple expressions.

Task No. 1

\[\left\( \begin(align)& 5x-4y=22 \\& 7x+4y=2 \\\end(align) \right.\]

Note that $y$ has a coefficient of $-4$ in the first equation, and $+4$ in the second. They are mutually opposite, so it is logical to assume that if we add them up, then in the resulting sum the “games” will be mutually destroyed. Add it up and get:

Let's solve the simplest construction:

Great, we found the "x". What should we do with it now? We have the right to substitute it into any of the equations. Let's substitute in the first:

\[-4y=12\left| :\left(-4 \right) \right.\]

Answer: $\left(2;-3 \right)$.

Problem No. 2

\[\left\( \begin(align)& -6x+y=21 \\& 6x-11y=-51 \\\end(align) \right.\]

The situation here is completely similar, only with “X’s”. Let's add them up:

We have the simplest linear equation, let's solve it:

Now let's find $x$:

Answer: $\left(-3;3 \right)$.

Important points

So, we have just solved two simple systems of linear equations using the addition method. Key points again:

If there are opposite coefficients for one of the variables, then it is necessary to add all the variables in the equation. In this case, one of them will be destroyed.
We substitute the found variable into any of the system equations to find the second one.
The final response record can be presented in different ways. For example, like this - $x=...,y=...$, or in the form of coordinates of points - $\left(...;... \right)$. The second option is preferable. The main thing to remember is that the first coordinate is $x$, and the second is $y$.
The rule of writing the answer in the form of point coordinates is not always applicable. For example, it cannot be used when the variables are not $x$ and $y$, but, for example, $a$ and $b$.

In the following problems we will consider the technique of subtraction when the coefficients are not opposite.

Solving easy problems using the subtraction method

Task No. 1

\[\left\( \begin(align)& 10x-3y=5 \\& -6x-3y=-27 \\\end(align) \right.\]

Note that there are no opposite coefficients here, but there are identical ones. Therefore, we subtract the second from the first equation:

Now we substitute the value $x$ into any of the system equations. Let's go first:

Answer: $\left(2;5\right)$.

Problem No. 2

\[\left\( \begin(align)& 5x+4y=-22 \\& 5x-2y=-4 \\\end(align) \right.\]

We again see the same coefficient of $5$ for $x$ in the first and second equation. Therefore, it is logical to assume that you need to subtract the second from the first equation:

We have calculated one variable. Now let's find the second one, for example, by substituting the value $y$ into the second construction:

Answer: $\left(-3;-2 \right)$.

Nuances of the solution

So what do we see? Essentially, the scheme is no different from the solution of previous systems. The only difference is that we do not add equations, but subtract them. We are doing algebraic subtraction.

In other words, as soon as you see a system consisting of two equations in two unknowns, the first thing you need to look at is the coefficients. If they are the same anywhere, the equations are subtracted, and if they are opposite, the addition method is used. This is always done so that one of them disappears, and in the final equation, which remains after subtraction, only one variable remains.

Of course, that's not all. Now we will consider systems in which the equations are generally inconsistent. Those. There are no variables in them that are either the same or opposite. In this case, to solve such systems, an additional technique is used, namely, multiplying each of the equations by a special coefficient. How to find it and how to solve such systems in general, we’ll talk about this now.

Solving problems by multiplying by a coefficient

Example #1

\[\left\( \begin(align)& 5x-9y=38 \\& 3x+2y=8 \\\end(align) \right.\]

We see that neither for $x$ nor for $y$ the coefficients are not only mutually opposite, but also in no way correlated with the other equation. These coefficients will not disappear in any way, even if we add or subtract the equations from each other. Therefore, it is necessary to apply multiplication. Let's try to get rid of the $y$ variable. To do this, we multiply the first equation by the coefficient of $y$ from the second equation, and the second equation by the coefficient of $y$ from the first equation, without touching the sign. We multiply and get a new system:

\[\left\( \begin(align)& 10x-18y=76 \\& 27x+18y=72 \\\end(align) \right.\]

Let's look at it: at $y$ the coefficients are opposite. In such a situation, it is necessary to use the addition method. Let's add:

Now we need to find $y$. To do this, substitute $x$ into the first expression:

\[-9y=18\left| :\left(-9 \right) \right.\]

Answer: $\left(4;-2 \right)$.

Example No. 2

\[\left\( \begin(align)& 11x+4y=-18 \\& 13x-6y=-32 \\\end(align) \right.\]

Again, the coefficients for none of the variables are consistent. Let's multiply by the coefficients of $y$:

\[\left\( \begin(align)& 11x+4y=-18\left| 6 \right. \\& 13x-6y=-32\left| 4 \right. \\\end(align) \right .\]

\[\left\( \begin(align)& 66x+24y=-108 \\& 52x-24y=-128 \\\end(align) \right.\]

Our new system is equivalent to the previous one, however, the coefficients of $y$ are mutually opposite, and therefore it is easy to apply the addition method here:

Now let's find $y$ by substituting $x$ into the first equation:

Answer: $\left(-2;1 \right)$.

Nuances of the solution

The key rule here is the following: we always multiply only by positive numbers- this will save you from stupid and offensive mistakes associated with changing signs. In general, the solution scheme is quite simple:

We look at the system and analyze each equation.
If we see that neither $y$ nor $x$ the coefficients are consistent, i.e. they are neither equal nor opposite, then we do the following: we select the variable that we need to get rid of, and then we look at the coefficients of these equations. If we multiply the first equation by the coefficient from the second, and the second, correspondingly, multiply by the coefficient from the first, then in the end we will get a system that is completely equivalent to the previous one, and the coefficients of $y$ will be consistent. All our actions or transformations are aimed only at getting one variable in one equation.
We find one variable.
We substitute the found variable into one of the two equations of the system and find the second.
We write the answer in the form of coordinates of points if we have variables $x$ and $y$.

But even such a simple algorithm has its own subtleties, for example, the coefficients of $x$ or $y$ can be fractions and other “ugly” numbers. We will now consider these cases separately, because in them you can act somewhat differently than according to the standard algorithm.

Solving problems with fractions

Example #1

\[\left\( \begin(align)& 4m-3n=32 \\& 0.8m+2.5n=-6 \\\end(align) \right.\]

First, notice that the second equation contains fractions. But note that you can divide $4$ by $0.8$. We will receive $5$. Let's multiply the second equation by $5$:

\[\left\( \begin(align)& 4m-3n=32 \\& 4m+12.5m=-30 \\\end(align) \right.\]

We subtract the equations from each other:

We found $n$, now let's count $m$:

Answer: $n=-4;m=5$

Example No. 2

\[\left\( \begin(align)& 2.5p+1.5k=-13\left| 4 \right. \\& 2p-5k=2\left| 5 \right. \\\end(align )\right.\]

Here, as in the previous system, there are fractional coefficients, but for none of the variables do the coefficients fit into each other an integer number of times. Therefore, we use the standard algorithm. Get rid of $p$:

\[\left\( \begin(align)& 5p+3k=-26 \\& 5p-12.5k=5 \\\end(align) \right.\]

We use the subtraction method:

Let's find $p$ by substituting $k$ into the second construction:

Answer: $p=-4;k=-2$.

Nuances of the solution

That's all optimization. In the first equation, we did not multiply by anything at all, but multiplied the second equation by $5$. As a result, we received a consistent and even identical equation for the first variable. In the second system, we followed a standard algorithm.

But how do you find the numbers by which to multiply equations? After all, if we multiply by fractions, we get new fractions. Therefore, the fractions must be multiplied by a number that would give a new integer, and after that the variables must be multiplied by coefficients, following the standard algorithm.

In conclusion, I would like to draw your attention to the format for recording the response. As I already said, since here we have not $x$ and $y$, but other values, we use a non-standard notation of the form:

Solving complex systems of equations

As a final note to today's video tutorial, let's look at a couple of really complex systems. Their complexity will consist in the fact that they will have variables on both the left and right. Therefore, to solve them we will have to apply preprocessing.

System No. 1

\[\left\( \begin(align)& 3\left(2x-y \right)+5=-2\left(x+3y \right)+4 \\& 6\left(y+1 \right )-1=5\left(2x-1 \right)+8 \\\end(align) \right.\]

Each equation carries a certain complexity. Therefore, let's treat each expression as with a regular linear construction.

In total, we get the final system, which is equivalent to the original one:

\[\left\( \begin(align)& 8x+3y=-1 \\& -10x+6y=-2 \\\end(align) \right.\]

Let's look at the coefficients of $y$: $3$ fits into $6$ twice, so let's multiply the first equation by $2$:

\[\left\( \begin(align)& 16x+6y=-2 \\& -10+6y=-2 \\\end(align) \right.\]

The coefficients of $y$ are now equal, so we subtract the second from the first equation: $$

Now let's find $y$:

Answer: $\left(0;-\frac(1)(3) \right)$

System No. 2

\[\left\( \begin(align)& 4\left(a-3b \right)-2a=3\left(b+4 \right)-11 \\& -3\left(b-2a \right )-12=2\left(a-5 \right)+b \\\end(align) \right.\]

Let's transform the first expression:

Let's deal with the second one:

\[-3\left(b-2a \right)-12=2\left(a-5 \right)+b\]

\[-3b+6a-12=2a-10+b\]

\[-3b+6a-2a-b=-10+12\]

In total, our initial system will take the following form:

\[\left\( \begin(align)& 2a-15b=1 \\& 4a-4b=2 \\\end(align) \right.\]

Looking at the coefficients of $a$, we see that the first equation needs to be multiplied by $2$:

\[\left\( \begin(align)& 4a-30b=2 \\& 4a-4b=2 \\\end(align) \right.\]

Subtract the second from the first construction:

Now let's find $a$:

Answer: $\left(a=\frac(1)(2);b=0 \right)$.

That's all. I hope this video tutorial will help you understand this difficult topic, namely solving systems of simple linear equations. There will be many more lessons on this topic: we will look at more complex examples, where there will be more variables, and the equations themselves will already be nonlinear. See you again!

A system of linear equations with two unknowns is two or more linear equations for which it is necessary to find all of them general solutions. We will consider systems of two linear equations in two unknowns. General form a system of two linear equations with two unknowns is presented in the figure below:

( a1*x + b1*y = c1,
( a2*x + b2*y = c2

Here x and y are unknown variables, a1, a2, b1, b2, c1, c2 are some real numbers. A solution to a system of two linear equations in two unknowns is a pair of numbers (x,y) such that if we substitute these numbers into the equations of the system, then each of the equations of the system turns into a true equality. There are several ways to solve a system of linear equations. Let's consider one of the ways to solve a system of linear equations, namely the addition method.

Algorithm for solving by addition method

An algorithm for solving a system of linear equations with two unknowns using the addition method.

1. If required, by means of equivalent transformations, equalize the coefficients of one of the unknown variables in both equations.

2. By adding or subtracting the resulting equations, obtain a linear equation with one unknown

3. Solve the resulting equation with one unknown and find one of the variables.

4. Substitute the resulting expression into any of the two equations of the system and solve this equation, thus obtaining the second variable.

5. Check the solution.

An example of a solution using the addition method

For greater clarity, let us solve the following system of linear equations with two unknowns using the addition method:

(3*x + 2*y = 10;
(5*x + 3*y = 12;

Since none of the variables have identical coefficients, we equalize the coefficients of the variable y. To do this, multiply the first equation by three, and the second equation by two.

(3*x+2*y=10 |*3
(5*x + 3*y = 12 |*2

We get the following system of equations:

(9*x+6*y = 30;
(10*x+6*y=24;

Now we subtract the first from the second equation. We present similar terms and solve the resulting linear equation.

10*x+6*y - (9*x+6*y) = 24-30; x=-6;

We substitute the resulting value into the first equation from our original system and solve the resulting equation.

(3*(-6) + 2*y =10;
(2*y=28; y =14;

The result is a pair of numbers x=6 and y=14. We are checking. Let's make a substitution.

(3*x + 2*y = 10;
(5*x + 3*y = 12;

{3*(-6) + 2*(14) = 10;
{5*(-6) + 3*(14) = 12;

{10 = 10;
{12=12;

As you can see, we got two correct equalities, therefore, we found the correct solution.