Problems solved using the equation: examples, explanation. Algebra Problems
Sooner or later, any schoolchild in the lessons of algebra will encounter problems solved by the equation. Initially, the appearance of letters instead of the usual numbers and actions with them confound even the most gifted, but if you look, everything is not as difficult as it seems at first glance.
Algorithm of the decision
Before turning to specific examples, it is necessary to understand the algorithm for solving problems using equations. In any equation there is an unknown, most often denoted by the letter X. Also in every problem there is what needs to be found, the same unknown. It is precisely this that must be designated as X. And then, following the condition of the problem, add, subtract, multiply and divide - perform any necessary actions.
After finding the unknown, it is imperative to perform the check to be sure that the problem is solved correctly. It is worth noting that children already in elementary school begin solving problems using equations.Examples of this are those tasks that need to be solved by segments, which are the most complete analogues of the letter unknowns.
The basis of the basics - the problem of the basket
So, let us try in practice to apply the solution of problems using equations, the explanation of the algorithm of which was given a little higher.
Given the problem: CThey made some baskets of apples. First, 3 baskets were sold, then 8 more baskets were finished. The result was 12 baskets. How many baskets of apples were originally harvested?
We begin the solution of the problem by designating the unknown — that is, the initial number of baskets — with the letter X. ), that is, X - 3 + 8 = 12. Having solved a simple equation, we get that X = 7. We will definitely perform the test, that is, we substitute the found number into the equation: 7 - 3 + 8 is indeed 12, that is, the problem is solved correctly.
Fastening: concert halls
Given the following task:two concert halls 450 seats. It is known that in one hall there are 4 times more seats than in the other. Need to know how many seats in each room.
In order to solve similar problems in algebra, we again need to apply the equation.We know that the sum of two numbers, one of which is 4 times the other, is 450. Let the number of seats in the smaller hall, the unknown, be equal to X, then the number of seats in the larger hall is 4 * X = 4X. Therefore, 450 = X + 4X = 5X. And then you need to solve the standard equation 450 = 5X, where X = 450/5 = 90, that is, in the smaller hall there are 90 seats, which means in the larger one - 90 * 4 = 360. To verify that the problem is solved correctly, you can check the inequality: 360 + 90 = 450, that is, the answer is correct.
But the problems solved by the equation can be more complicated. For example,There are three shelves with books. There are 8 more books on the first shelf than on the second, and 3 times more on the third than the second, and the number of books on the first and third shelves is equal. How many books are on each shelf?
It is clear that you need to push off from the second shelf, which is found in both conditions. If we designate the number of books on it for X, then on the first shelf X + 8 books, and on the third - X * 3 books, while X + 8 = 3X. Solving the equation, we get X = 4. We perform the test, substituting the unknown into equality: 4 + 8 is really equal to 3 * 4, that is, the problem is solved correctly.
Practice further: beavers
As you can see, solving problems using an equation is much easier than it seems at first glance. We will fix the skills of working with equations by another task.The first beaver gnawed some trees in a day. The second beaver gnawed 6 times more. The third beaver gnawed 2 times more trees than the first, but 3 times less than the second. How many trees did each beaver bite?
The task is not as complicated as it seems at first glance. First we find the unknown - in this problem, this is the number of trees gnawed by the first beaver. Consequently, the second beaver destroyed 6 * X trees, and the third - 2 * X, and this number is 3 times less than 6 * X. We make the equation: 6X = 3 * 2X. Having solved it, we get that the first beaver gnawed on just one tree, then the second - 6, and the third - 2. Substituting the numbers into the equation, we understand that the problem is solved correctly.
We correlate the equations and conditions
If you are told: "For each problem, select the appropriate equation," - do not worry - this is entirely real.
The following equations are given:
- 6 + X = 2X;
- 6 = 2X;
- 2 + X = 6.
The conditions of the tasks are as follows:
- The boy had 6 apples, and the girl was two times less, how many apples did the girl have?
- On the table are pens and pencils, it is known that there are 6 pens on the table, and 2 pencils less, how many pens and how many pencils on the table?
- Vanya has six coins more than Tanya, and Tanya has two times less than Ani, how many coins does each child have, if Vanya and Ani have the same amount of coins?
We make equations for each of the problems.
- In the first case, we do not know the number of apples in a girl, that is, it is equal to X, we know that X is 2 times less than 6, that is, 6 = 2X, therefore, equation No. 2 fits this condition.
- In the second case, X indicates the number of pencils, then the number of X + 2 pens, but we also know that there are 6 pens, that is, X + 2 = 6, which means the third equation fits here.
- As for the last task, at number 3, the number of Tannins, which occurs in two conditions, is the unknown unknown, then Vanya has 6 + X coins, and Ani has 2X coins, that is, 6 + X = 2X - it is obvious that first equation.
If you have problems to be solved using an equation, to which you need to find the appropriate equality, then make up an equation for each of the problems, and then correlate what you have done to these equations.
Complicate: the system of equations - candy
The next stage in the application of letter equality in algebra is problems solved by a system of equations. They have two unknowns, and one of them is expressed in terms of the other, based on the available data.It is known that Pasha and Katie together 20 candies. It is also known that if Pasha had 2 more candies, he would have 15 candies, how many candies each?
In this case, we do not know the number of Katy's candies, nor the number of Sasha's candies, therefore, we have two unknowns, X and Y, respectively. At the same time, we know that Y + 2 = 15.
Making a system, we get two equations:
- X + Y = 20;
- Y + 2 = 15.
And then we act according to the rules of solving systems: we derive Y from the second equation, getting Y = 15 - 2, and then we substitute it into the first, that is, X + Y = X + (15 - 2) = 20. Having solved the equation, we get X = 7, then Y = 20 - 7 = 13. Check the correctness of the solution, substituting Y in the second equation: 13 + 2 is really equal to 15, that is, Katya has 7 candies, and for Pasha - 13.
Even more difficult: quadratic equations and land
There are also problems on algebra that can be solved by a quadratic equation. There is nothing difficult in them, just the standard system is transformed into a quadratic equation during the solution. For example,Given a plot of land of 6 hectares (60000 square meters), the fence enclosing it has a length of 1000 meters. What are the length and width of the plot?
We make up the equation. The length of the fence is the perimeter of the site, therefore, if the length is denoted by X, and the width is Y, then 1000 = 2 * (X + Y). The area is the same, that is X * Y = 60000. From the first equation we derive X = 500 - Y. Substituting it into the second equation, we get (500 - Y) * Y = 60000, that is, 500Y - Y2= 60000.Having solved the equation, we get sides equal to 200 and 300 meters - the quadratic equation has two roots, one of which is often not suitable for the condition, for example, is negative, whereas the answer must be a natural number, therefore it is necessary to check.
Repeat: trees in the garden
Fixing the topic, we solve another problem.In the garden there are several apples, 6 pears and several cherry trees. It is known that the total number of trees is 5 times more than the number of apple trees, while there are 2 times more cherry trees than apple trees. How many trees are of each kind in the garden and how many in the garden are all the trees?
For the unknown X, as is probably already clear, we denote apple trees, through which we can express the other quantities. It is known that Y = 2X, and Y + X + 6 = 5X. Substituting Y from the first equation, we obtain the equality 2X + X + 6 = 5X, whence X = 3, therefore in the garden Y = 3 * 2 = 6 cherry trees. We check and answer the second question, adding the resulting values: 3 + 6 + 6 = 3 * 5, that is, the problem is solved correctly.
Control: the sum of numbers
Solving problems using an equation is far from being as complicated as it seems at first glance. The main thing is not to make a mistake in choosing the unknown and, more importantly, to express it correctly, especially if we are talking about a system of equations.In conclusion, the last problem is given, much more involved than the ones presented above.
The sum of three numbers is 40. It is known that X = 2Y + 3Z, and Y = Z - 2/3. What are X, Y and Z equal to?
So let's start with getting rid of the first unknown. Instead of X, we substitute the corresponding expression in the equality, we get 2Y + 3Z + Z + Y = 3Y + 4Z = 40. Next, we also replace the known Y, getting the equality 3Z - 2 + 4Z = 40, whence Z = 6. Returning to Y, we find that it is equal to 5.2, and X, in turn, is equal to 18. With the help of verification we are convinced of the truth of the expression, therefore the problem is solved correctly.
So, what are the problems solved by the equation? Are they as scary as it seems at first glance? In no case! With due diligence to understand them is not difficult. And once you understand the algorithm, in the future you will be able to click on similar puzzles, even the most intricate, like seeds. The main thing is attentiveness, it is she who will help to correctly determine the unknown and, by solving sometimes a set of equations, find the answer.