Instructions
Perhaps the most obvious point here is, of course. Numerical fractions do not pose any danger (fractional equations, where all denominators contain only numbers, will generally be linear), but if there is a variable in the denominator, then this must be taken into account and written down. Firstly, it is that x, which turns the denominator to 0, cannot be, and in general it is necessary to separately state the fact that x cannot be equal to this number. Even if you succeed that when substituting into the numerator, everything converges perfectly and satisfies the conditions. Secondly, we cannot multiply either side of the equation by , which is equal to zero.
After this, such an equation is reduced to moving all its terms to the left side so that 0 remains on the right.
It is necessary to bring all terms to a common denominator, multiplying, where necessary, the numerators by the missing expressions.
Next, we solve the usual equation written in the numerator. We can take common factors out of brackets, use abbreviated multiplication, bring similar ones, calculate the roots of a quadratic equation through the discriminant, etc.
The result should be a factorization in the form of a product of brackets (x-(i-th root)). This may also include polynomials that do not have roots, for example, quadratic trinomial with a discriminant less than zero (if, of course, the problem contains only real roots, as most often happens).
It is imperative to factorize the denominator and find the parentheses already contained in the numerator. If the denominator contains expressions like (x-(number)), then it is better not to multiply the parentheses in it directly when reducing to a common denominator, but to leave them as a product of the original simple expressions.
Identical parentheses in the numerator and denominator can be shortened by first writing down, as mentioned above, the conditions on x.
The answer is written in curly brackets, as a set of x values, or simply as an enumeration: x1=..., x2=..., etc.
Sources:
- Fractional rational equations
Something you can’t do without in physics, mathematics, chemistry. Least. Let's learn the basics of solving them.
Instructions
The most general and simple classification can be divided according to the number of variables they contain and the degrees in which these variables stand.
Solve the equation with all its roots or prove that there are none.
Any equation has no more than P roots, where P is the maximum of a given equation.
But some of these roots may coincide. So, for example, the equation x^2+2*x+1=0, where ^ is the icon for exponentiation, is folded into the square of the expression (x+1), that is, into the product of two identical brackets, each of which gives x=- 1 as a solution.
If there is only one unknown in an equation, this means that you will be able to explicitly find its roots (real or complex).
For this, you will most likely need various transformations: abbreviated multiplication, calculation of the discriminant and roots of a quadratic equation, transfer of terms from one part to another, reduction to a common denominator, multiplication of both parts of the equation by the same expression, by a square, etc.
Transformations that do not affect the roots of the equation are identical. They are used to simplify the process of solving an equation.
You can also use instead of traditional analytical graphical method and write this equation in the form, then carry out its study.
If there is more than one unknown in an equation, then you will only be able to express one of them in terms of the other, thereby showing a set of solutions. These are, for example, equations with parameters in which there is an unknown x and a parameter a. To solve a parametric equation means for all a to express x in terms of a, that is, to consider all possible cases.
If the equation contains derivatives or differentials of unknowns (see picture), congratulations, this differential equation, and here you can’t do without higher mathematics).
Sources:
- Identity transformations
To solve the problem with in fractions, you need to learn how to deal with them arithmetic operations. They can be decimal, but are most often used natural fractions with a numerator and denominator. Only after this can you move on to solving mathematical problems with fractional quantities.
You will need
- - calculator;
- - knowledge of the properties of fractions;
- - ability to perform operations with fractions.
Instructions
A fraction is a notation for dividing one number by another. Often this cannot be done completely, which is why this action is left unfinished. The number that is divisible (it appears above or before the fraction sign) is called the numerator, and the second number (below or after the fraction sign) is called the denominator. If the numerator is greater than the denominator, the fraction is called an improper fraction, and a whole part can be separated from it. If the numerator is less than the denominator, then such a fraction is called proper, and its integer part is equal to 0.
Tasks are divided into several types. Determine which of them the task belongs to. The simplest option– finding the fraction of a number expressed as a fraction. To solve this problem, just multiply this number by a fraction. For example, 8 tons of potatoes were delivered. In the first week, 3/4 of its total was sold. How many potatoes are left? To solve this problem, multiply the number 8 by 3/4. It turns out 8∙3/4=6 t.
If you need to find a number by its part, multiply the known part of the number by the inverse fraction of the one that shows what the share of this part is in the number. For example, 8 of them make up 1/3 of the total number of students. How many in ? Since 8 people is a part that represents 1/3 of the total, then find the reciprocal fraction, which is 3/1 or just 3. Then to get the number of students in the class 8∙3=24 students.
When you need to find what part of a number one number is from another, divide the number that represents the part by the one that is the whole. For example, if the distance is 300 km, and the car has traveled 200 km, what part of the total distance will this be? Divide part of the path 200 by full path 300, after reducing the fraction you will get the result. 200/300=2/3.
To find an unknown fraction of a number when there is a known one, take the whole number as a conventional unit and subtract the known fraction from it. For example, if 4/7 of the lesson has already passed, is there still time left? Take the entire lesson as a unit and subtract 4/7 from it. Get 1-4/7=7/7-4/7=3/7.
Solving equations with fractions Let's look at examples. The examples are simple and illustrative. With their help, you will be able to understand in the most understandable way.
For example, you need to solve the simple equation x/b + c = d.
An equation of this type is called linear, because The denominator contains only numbers.
The solution is performed by multiplying both sides of the equation by b, then the equation takes the form x = b*(d – c), i.e. the denominator of the fraction on the left side cancels.
For example, how to solve fractional equation:
x/5+4=9
We multiply both sides by 5. We get:
x+20=45
x=45-20=25
Another example when the unknown is in the denominator:
Equations of this type are called fractional-rational or simply fractional.
We would solve a fractional equation by getting rid of fractions, after which this equation, most often, turns into a linear or quadratic equation, which can be solved in the usual way. You just need to consider the following points:
- the value of a variable that turns the denominator to 0 cannot be a root;
- You cannot divide or multiply an equation by the expression =0.
This is where the concept of the region of permissible values (ADV) comes into force - these are the values of the roots of the equation for which the equation makes sense.
Thus, when solving the equation, it is necessary to find the roots, and then check them for compliance with the ODZ. Those roots that do not correspond to our ODZ are excluded from the answer.
For example, you need to solve a fractional equation:
Based on the above rule, x cannot be = 0, i.e. ODZ in in this case: x – any value other than zero.
We get rid of the denominator by multiplying all terms of the equation by x
And we solve the usual equation
5x – 2x = 1
3x = 1
x = 1/3
Answer: x = 1/3
Let's solve a more complicated equation:
ODZ is also present here: x -2.
When solving this equation, we will not move everything to one side and bring the fractions to a common denominator. We will immediately multiply both sides of the equation by an expression that will cancel out all the denominators at once.
To reduce the denominators, you need to multiply the left side by x+2, and the right side by 2. This means that both sides of the equation must be multiplied by 2(x+2):
This is the most common multiplication of fractions, which we have already discussed above.
Let's write the same equation, but slightly differently
The left side is reduced by (x+2), and the right by 2. After the reduction, we obtain the usual linear equation:
x = 4 – 2 = 2, which corresponds to our ODZ
Answer: x = 2.
Solving equations with fractions not as difficult as it might seem. In this article we have shown this with examples. If you have any difficulties with how to solve equations with fractions, then unsubscribe in the comments.
The lowest common denominator is used to simplify this equation. This method is used when you cannot write a given equation with one rational expression on each side of the equation (and use the crisscross method of multiplication). This method is used when you are given a rational equation with 3 or more fractions (in the case of two fractions, it is better to use criss-cross multiplication).
Find the lowest common denominator of the fractions (or least common multiple). NOZ is the smallest number that is evenly divisible by each denominator.
- Sometimes NPD is an obvious number. For example, if given the equation: x/3 + 1/2 = (3x +1)/6, then it is obvious that the least common multiple of the numbers 3, 2 and 6 is 6.
- If the NCD is not obvious, write down the multiples of the largest denominator and find among them one that will be a multiple of the other denominators. Often the NOD can be found by simply multiplying two denominators. For example, if the equation is given x/8 + 2/6 = (x - 3)/9, then NOS = 8*9 = 72.
- If one or more denominators contain a variable, the process becomes somewhat more complicated (but not impossible). In this case, the NOC is an expression (containing a variable) that is divided by each denominator. For example, in the equation 5/(x-1) = 1/x + 2/(3x) NOZ = 3x(x-1), because this expression is divided by each denominator: 3x(x-1)/(x-1 ) = 3x; 3x(x-1)/3x = (x-1); 3x(x-1)/x = 3(x-1).
Multiply both the numerator and denominator of each fraction by a number equal to the result of dividing the NOC by the corresponding denominator of each fraction. Since you are multiplying both the numerator and denominator by the same number, you are effectively multiplying the fraction by 1 (for example, 2/2 = 1 or 3/3 = 1).
- So in our example, multiply x/3 by 2/2 to get 2x/6, and 1/2 multiply by 3/3 to get 3/6 (the fraction 3x +1/6 does not need to be multiplied because it the denominator is 6).
- Proceed similarly when the variable is in the denominator. In our second example, NOZ = 3x(x-1), so multiply 5/(x-1) by (3x)/(3x) to get 5(3x)/(3x)(x-1); 1/x multiplied by 3(x-1)/3(x-1) and you get 3(x-1)/3x(x-1); 2/(3x) multiplied by (x-1)/(x-1) and you get 2(x-1)/3x(x-1).
Find x. Now that you have reduced the fractions to a common denominator, you can get rid of the denominator. To do this, multiply each side of the equation by the common denominator. Then solve the resulting equation, that is, find “x”. To do this, isolate the variable on one side of the equation.
- In our example: 2x/6 + 3/6 = (3x +1)/6. You can add 2 fractions with same denominator, so write the equation as: (2x+3)/6=(3x+1)/6. Multiply both sides of the equation by 6 and get rid of the denominators: 2x+3 = 3x +1. Solve and get x = 2.
- In our second example (with a variable in the denominator), the equation looks like (after reduction to a common denominator): 5(3x)/(3x)(x-1) = 3(x-1)/3x(x-1) + 2 (x-1)/3x(x-1). By multiplying both sides of the equation by N3, you get rid of the denominator and get: 5(3x) = 3(x-1) + 2(x-1), or 15x = 3x - 3 + 2x -2, or 15x = x - 5 Solve and get: x = -5/14.
Solving equations with fractions Let's look at examples. The examples are simple and illustrative. With their help, you will be able to understand in the most understandable way.
For example, you need to solve the simple equation x/b + c = d.
An equation of this type is called linear, because The denominator contains only numbers.
The solution is performed by multiplying both sides of the equation by b, then the equation takes the form x = b*(d – c), i.e. the denominator of the fraction on the left side cancels.
For example, how to solve a fractional equation:
x/5+4=9
We multiply both sides by 5. We get:
x+20=45
x=45-20=25
Another example when the unknown is in the denominator:
Equations of this type are called fractional-rational or simply fractional.
We would solve a fractional equation by getting rid of fractions, after which this equation, most often, turns into a linear or quadratic equation, which is solved in the usual way. You just need to consider the following points:
- the value of a variable that turns the denominator to 0 cannot be a root;
- You cannot divide or multiply an equation by the expression =0.
This is where the concept of the region of permissible values (ADV) comes into force - these are the values of the roots of the equation for which the equation makes sense.
Thus, when solving the equation, it is necessary to find the roots, and then check them for compliance with the ODZ. Those roots that do not correspond to our ODZ are excluded from the answer.
For example, you need to solve a fractional equation:
Based on the above rule, x cannot be = 0, i.e. ODZ in this case: x – any value other than zero.
We get rid of the denominator by multiplying all terms of the equation by x
And we solve the usual equation
5x – 2x = 1
3x = 1
x = 1/3
Answer: x = 1/3
Let's solve a more complicated equation:
ODZ is also present here: x -2.
When solving this equation, we will not move everything to one side and bring the fractions to a common denominator. We will immediately multiply both sides of the equation by an expression that will cancel out all the denominators at once.
To reduce the denominators, you need to multiply the left side by x+2, and the right side by 2. This means that both sides of the equation must be multiplied by 2(x+2):
This is the most common multiplication of fractions, which we have already discussed above.
Let's write the same equation, but slightly differently
The left side is reduced by (x+2), and the right by 2. After the reduction, we obtain the usual linear equation:
x = 4 – 2 = 2, which corresponds to our ODZ
Answer: x = 2.
Solving equations with fractions not as difficult as it might seem. In this article we have shown this with examples. If you have any difficulties with how to solve equations with fractions, then unsubscribe in the comments.