How to solve fractional equations, how to solve fractional equations

Fractional equation solution:

1) Remove the denominator.

Both sides of the equation are multiplied by the simplest common denominator (simplest common denominator: the coefficient is taken as the least common multiple; The letter that appears is taken to the highest power; The occurring factor takes the highest power), and the fractional equation is reduced to an integral equation; If you encounter opposite numbers, don't forget to change the number.

2) Root inspection. After finding the value of the unknown, it is necessary to check the root, because in the process of converting the fractional equation into an integral equation, the range of the value of the unknown is expanded, and the root may be increased.

If the simplest common denominator is equal to 0, this root is the additional root of the original equation. Otherwise this root is the root of the original fractional equation. If the solved root is an additional root, then there is no solution to the original equation.

If the fraction itself is reduced, it should also be substituted for the original equation test.

When solving a problem with a column fractional equation, it is necessary to check not only whether the solution satisfies the equation, but also whether it conforms to the meaning of the problem.

In general, when solving a fractional equation, the solution of the integral equation obtained after removing the denominator may make the denominator in the original equation zero, so the solution of the integral equation should be substituted into the simplest common denominator, and if the value of the simplest common denominator is not zero, it is the solution of the equation.

Note: 1) When removing the denominator, do not omit to multiply the integer term.

2) The root is the root of the integral equation formed by removing the denominator from the fractional equation, but it is not the solution of the original fractional equation.

3) Root increment so that the simplest common denominator is equal to 0.

Fractional Equation Concept:

A fractional equation is a type of equation in which the denominator contains unknowns (rational) is called a fractional equation. A rational equation with at least one denominator on either side of the equal sign and an unknown number is called a fractional equation.

The equation is not very clear and does not distinguish between the last two items on the right.

A fractional equation is a digging equation with a fraction in the equation, the fractions a b, a and b are all integers, and the denominator b contains the letters, b ≠ 0, for example: 8 x = 4. The solution method of fractional equation is to remove the denominator first, then remove the parentheses, then move the terms, merge the similar terms, and the coefficient is turned into 1, and finally the judgment is called.

The first step is to remove the denominator, multiply the simplest common denominator of each denominator on both sides of the equation, and solve 3 (x+1)=5 (x+3). Multiply (x+1) (x+3) to remove the denominator.

The second step is to remove the brackets and multiply the coefficients by the numbers in the brackets.

In the third step, the equation containing the unknown number moves to the left of the equation and the constant moves to the right of the equation.

The fourth step is to merge similar items.

In the fifth step, the coefficient is reduced to 1, and the basic property of the equation is to multiply or divide by a number at the same time, and the equation does not change, just like the balance. Divide by -2 here.

The sixth step is to test whether the solution of the equation is substituted into the fractional equation and whether it is correct.

The method of solving a fractional equation is to remove the denominator, multiply the simplest common denominator of each denominator on both sides of the equation, remove the parentheses, shift the terms, move the formula containing unknown numbers to the left side of the equation, move the constant to the right side of the equation, merge the similar terms, reduce the coefficient to 1, substitute the solution of the equation into the fractional equation, and check whether it is correct.

1. Remove the denominator, multiply the simplest common denominator of each denominator on both sides of the equation, and solve 3 (x+1)=5 (x+3). Multiply (x+1) (x+3) to remove the denominator.

2. Remove the brackets, and multiply the coefficients by the number in the brackets.

3. Shift terms, the formula containing unknown numbers moves to the left of the equation, and the constants move to the right side of the equation.

4. Merge similar items.

5. The coefficient is 1, and the basic property of the equation is to multiply or divide by a number at the same time, and the equation remains unchanged, which is the same as the balance. Divide by -2 here.

6. Test, substitute the solution of the equation into the fractional equation to check whether it is correct.

The difference between a fractional equation without a solution and an increase in roots:

1. The root of the fractional equation is required to be the root of the transformed integral equation first.

2. The method of solving fractions is to convert the fractional equation into an integral equation by removing the denominator.

3. Therefore, it is concluded that the root of the fractional equation must be the root of the simplified integral equation, and the root of the simplified integral equation is not necessarily the root of the fractional equation, but may be the root of the addition of the fractional equation, and the fractional equation has no solution, that is, the simplified integral equation has no solution.

4. Substituting the root obtained by the integral equation into the fractional equation, if the denominator in the fractional equation is not 0, then the root obtained is the root of the fractional equation, otherwise it is the root of the fractional equation.

5. Verify whether the root obtained by the integral equation is the root of the fractional equation.

The basic idea of solving fractional equations is to convert fractional equations into integral equations, and the specific method is to "remove the denominator", that is, the simplest common denominator is multiplied on both sides of the equation, which is also the general idea and practice of solving fractional equations. Example: (1) x (x+1) = 2x (3x+3)+1

Multiply both sides by 3 (x+1).

3x=2x+(3x+3)

3x=5x+3

2x=-3x=-3/2

Fractional equations to be tested.

After testing, x=-3 2 is the solution of the equation.

2）2/（x-1）=4/（x^2-1）

Multiply (x+1) (x-1).

2(x+1)=4

2x+2=4

2x=2x=1

Fractional equations to be tested.

Bring the talking branch roll x=1 into the original equation, so that the denominator is 0, which is the root increase. Therefore, the original recipe contains a surplus of 2 x-1 = 4 x 2-1

No solution. Be sure to check! Test format: put x=a

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.

The steps of the basis or solution of the fractional equation are to remove the denominator, multiply the simplest common denominator of each denominator on both sides of the equation, remove the parentheses, shift the term, move the formula containing unknown numbers to the left side of the equation, move the constant to the right side of the equation, merge the similar terms, turn the coefficient into 1, substitute the operation-only solution of the equation into the fractional equation, and test whether it is correct.

Fractional equation is a kind of equation, which refers to a rational equation with unknown numbers or integers with unknown numbers in the denominator, and this part of the knowledge belongs to elementary mathematical knowledge. An equation is an equation that contains unknowns. It is an equation that expresses the equality relationship between two mathematical formulas, such as two numbers, functions, quantities, and operations.

Steps to solve a fractional equation:

1. Remove the denominator, multiply the left and right sides of the equation by the simplest common denominator;

2. Solving integer equations;

3. Check whether the root of the equation is an additional root;

4. Conclusion.

The denominator is divided to find the least common multiple of the denominator, and then multiply the least common multiple on both sides of the equation to simplify it.

General steps for solving fractional equations:

1. Multiply the simplest common denominator on both sides of the equation, reduce the denominator, and form an integer equation.

2.Solve this integer equation.

3. Substitute the solution of the integer equation into the simplest common denominator, if the value of the simplest common denominator is not 0, then the solution of the integer equation is the solution of the original fractional equation; Otherwise, this solution is not the solution of the original fractional equation and must be discarded.

4. Write the root of the original equation.