To judge whether a formula is a fraction, the key is to satisfy:
The denominator of (1) score must contain letters.
(2) The value of denominator cannot be zero. If the denominator is zero, the score is meaningless.
Because letters can represent different numbers, fractions are more common than fractions.
Algebraic expressions and fractions are collectively called rational forms.
A formula with a root sign and letters below it is called an irrational formula.
Infinitely cyclic decimals are also irrational numbers.
Irrational and rational expressions are collectively called algebraic expressions.
algorithm
1. About integral:
The process of reducing the common factor of the numerator and denominator of a fraction is called reduction.
2. The multiplication rule of fractions:
Multiply two fractions, using the product of molecules as the numerator of the product and the product of denominator as the denominator of the product.
Invert the numerator and denominator of the divisor (the reciprocal of the divisor), then multiply the divisor to divide the two fractions.
3. The law of addition and subtraction of fractions:
Add and subtract fractions with the same denominator and numerator with the same denominator.
4. The law of addition and subtraction of fractions with different denominators:
Fractions with different denominators are added and subtracted, first divided into fractions with the same denominator, and then calculated according to the addition and subtraction law of fractions with the same denominator.
Remarks: Fractions with different denominators can be converted into fractions with the same denominator. This process is called general division. For example, 3/2 and 2/3 can be changed to 9/6 and 4/6. That is 3*3/2*3, 2*2/3*2.
Basic attribute
The basic property of 1. Fraction: the numerator and denominator of the fraction are multiplied (or divided) by the same algebraic expression that is not zero at the same time, and the value of the fraction remains unchanged. The formula is: A/B=(A*C)/(B*C), a/b = (a ÷ c)/(b ÷ c) (a, B and C are algebraic expressions, and B, C≠0).
2. Simplification: The common factor of the numerator and denominator of a fraction is simplified, which is called the simplification of the fraction. The key to reduction is to determine the common factor of numerator and denominator in fractions.
3. Step of score reduction:
(1) If the numerator and denominator of a fraction are both monomials or products of several factors, their common factors are removed.
(2) The numerator and denominator of a fraction are polynomials, which are decomposed into factors respectively, and then the common factor is removed.
Note: extraction method of common factor: coefficient takes the greatest common divisor of numerator denominator coefficient, letter takes the letter of numerator denominator * * *, and index takes the smallest index of common * * * letter as their common factor.
Simplest fraction: When the numerator and denominator of a fraction have no common factor, the fraction is called simplest fraction. When simplifying, a fraction is generally simplified to the simplest fraction.
5. According to the basic properties of fractions, fractions with different denominators can be divided equally, so that the denominators of several fractions are the same; Similarly, according to the basic properties of fractions, fractions can be similarly deformed, so that the denominators of several fractions with different denominators are the same, while the values of fractions remain unchanged.
6. Total score: It is called the total score of a score when several scores with different denominators are changed into scores with the same denominator equal to the original score.
7. The general steps of the score:
Find the simplest common denominator of all fractions first, and then change the denominator of all fractions into the simplest common denominator. At the same time, each fraction enlarges its own molecule according to the multiple of the denominator.
Note: How to determine the simplest common denominator:
The coefficient is the product of the least common multiple of each factor coefficient, the highest power of the same letter and the power of a single letter.
Note: (1) reduction and general division are based on the basic properties of fractions.
(2) Fractions and general fractions are reciprocal operations.
arithmetic
1. Addition and subtraction of the same denominator fraction: addition and subtraction of the same denominator fraction and addition and subtraction of the same denominator numerator. Expressed in letters: A/CB/C = (AB)/C.
2. Addition and subtraction of fractions with different denominators: add and subtract fractions with different denominators, first divide them into fractions with the same denominator, and then calculate them according to the addition and subtraction rules of fractions with the same denominator. Expressed in letters: a/b c/d = (ad CB)/BD.
3. Multiplication rule of fractions: two fractions are multiplied, the product of numerator multiplication is the numerator of product, and the product of denominator multiplication is the denominator of product. Expressed in letters: a/b * c/d=ac/bd.
4. The law of fractional division:
(1). Divide two fractions, invert the numerator and denominator of the divisor, and then multiply by the divisor. a/b÷c/d=ad/bc .
(2) dividing by a fraction is equal to multiplying the reciprocal of this fraction: a/b ÷ c/d = a/b * d/c.
5. Power law: numerator multiplied by numerator and denominator multiplied by denominator can be simplified to the simplest.
fractional equation
Equations with unknowns in the denominator are called fractional equations.
fractional equation
① Denominator {Both sides of the equation are multiplied by the simplest common denominator at the same time (the simplest common denominator: ① The coefficient is the least common multiple; (2) the letters that appear occupy the highest power; (3) the factors that appear take the highest power), and the fractional equation is transformed into an integral equation; If you encounter the opposite number. Don't forget to change the symbol.
(2) According to the steps of solving the integral equation (shift the term, remove the brackets if there are brackets, pay attention to the sign change, merge similar terms, and convert them into 1), and find out the unknown value;
③ Root test (root test is needed after finding the value of the unknown quantity, because in the process of transforming the fractional equation into the whole equation, the range of the unknown quantity is expanded, which may lead to the increase of roots).
Generally speaking, to test the root, we only need to substitute the root of the whole equation into the simplest common denominator. If the simplest common denominator is equal to 0, this root is an increased root, otherwise it is the root of the original fractional equation. If the root of the solution is an increasing root, the original equation has no solution.
If the score itself is divided, the score should be tested.
Problem solving steps
The general steps of solving application problems with fractional equations are as follows:
(1) Setting the unknown: if the unknown required in the topic is directly represented by letters, it is called directly setting the unknown, otherwise it is called indirectly setting the unknown;
(2) Column Algebraic Formula: use algebraic formula containing unknowns to express the relevant quantities in the topic, and make schematic diagrams or tables when necessary to help straighten out the relationship between quantities;
(3) listing equations: listing equations according to the obvious or implied equation relationship in the topic;
(4) Solve the equation and test it;
(5) write the answer.
When solving an application problem with a column fraction equation, it is necessary to check whether the solution satisfies the equation and whether the solution satisfies the meaning of the problem.
Generally speaking, when solving a fractional equation, the denominator of the whole equation may be zero after removing the denominator, so we should substitute the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not zero, it is the solution of the equation.
Application example
Example 1: Solve the equation (1) x/(x+1) = 2x/(3x+3)+1.
Multiply both sides by 3(x+ 1) and divide by the denominator.
3x=2x+(3x+3)
3x=5x+3
2x=-3
∴x=-3/2
X=-3/2 is the solution of the original equation.
(2)2/(x- 1)=4/(x^2- 1)
Multiply both sides by (x+ 1)(x- 1) and divide by the denominator.
2(x+ 1)=4
2x+2=4
2x=2
∴x= 1
Test: Bring x= 1 into the original equation and make the denominator 0, that is, increase the root.
So the original equation 2/(x-1) = 4/(x 2-1) has no solution.
(3)2x-3+ 1/(x-5)= x+2+ 1/(x-5)
Subtract 1/(x-5) from both sides to get x=5.
Substituting into the original equation to make the denominator 0, then x=5 is the increase of the root.
So the equation has no solution!
Test: bring x=a into the simplest common denominator. If x=a makes the simplest common denominator 0, then a is the root of the original equation. If x=a makes the simplest common denominator not zero, then a is the root of the original equation.
Induction: The basic idea of solving the fractional equation is to turn the fractional equation into an integral equation, and the specific method is to "remove the denominator", that is, both sides of the equation are multiplied by the simplest common denominator, which is also the general idea and method of solving the fractional equation. Test format: bring x=a into the simplest common denominator. If x=a makes the simplest common denominator 0, then a is the root of the original equation. If x=a makes the simplest common denominator not zero, then a is the root of the original equation.
Of course, we can judge whether there is a solution by experience. If there is a solution, substitute all denominators for calculation; If there is no solution, substitute it into the denominator without solution.
Example 2. (20 10 Shaoyang, Hunan) Xiaoming left home to watch a football match at the stadium. When he entered the stadium, he found that the ticket was still at home. At this time, there are still 45 minutes before the game, and he immediately walks home (at a constant speed) to get the ticket. It took him 2 minutes to get a ticket at home. As soon as he got the ticket, he went to the stadium by bike (at a constant speed). It is understood that it takes Xiaoming 20 minutes less to ride home from home than to walk home from the gym, and the speed of cycling is three times that of walking.
(1) What is Xiao Ming's walking speed (unit: m/min)?
(2) Can Xiao Ming get to the gym before the game starts?
Analysis (1) Suppose the walking speed is x m/min, then the cycling speed is 3x m/min.
(2400╱x)-(2400╱3x)=20.
X=80，3x=240。
X=80 is the root of the original equation.
Xiaoming walks at a speed of 80 meters per minute.
(2) The total time for going home to collect tickets is:
(2400 x)+(2400 3x)+2 = 42 minutes.
So that I can get to the stadium before the game starts.
Perhaps these landlords can find some examples to see if they will ask questions again.