Primary school is mainly about solving, while middle school is mainly about learning, and solving equations according to conditions. Therefore, it can be said that the application problem involving linear equations of one variable is the first difficulty in learning equations in middle schools. Let's first solve the first problem that may be left over from junior high school-the linear equation of one yuan.

What is a linear equation with one variable?

We call an equation with unknowns an equation, so what is a linear equation? There is only one unknown (element) X, and the exponent of the unknown X is 1 (degree). Such an equation is called a linear equation with one variable.

Standard form of one-dimensional linear equation;

Ax+b=0(x is unknown, a and b are known numbers, a≠0).

One-dimensional linear equation must meet four conditions at the same time:

(1) It is an equation;

(2) There are no unknowns in the denominator;

(3) The highest unknown term is1;

(4) The coefficient of the term with unknown number is not 0.

(1) denominator: the denominator is multiplied by the least common multiple on both sides of the equation.

(2) Bracket removal: according to the rules of bracket removal and distribution.

(3) Shift the term: move the term containing the unknown to one side of the equation, and all the other terms will be moved to the other side of the equation, and the shift term should be signed.

(4) Merge: transform the equation into ax = b (a≠0).

(5) Coefficient to 1: divide both sides of the equation by the unknown coefficient A to get the solution of the equation.

The value of the unknown quantity that makes the left and right sides of the equal sign in the equation equal is called the solution of the equation.

Note: the solution of the equation and the solution of the equation are different concepts. The solution of the equation is essentially the result of the solution, which is a numerical value (or several numerical values), and the meaning of solving the equation refers to the process of finding the solution of the equation or judging that the equation has no solution.

Solving a linear equation with one variable is actually solving it by using the properties of equality.

● Properties of the equation

When the same number (or formula) is added (or subtracted) on both sides, the equation (1) still holds.

Expressed in the form of a formula: if a=b, then a c = b c

(2) When both sides of the equation are multiplied by the same number or divided by the same number that is not 0, the equation still holds.

Expressed in the form of a formula: if a=b, then ac = bc If a=b(c≠0), then A/C = B/C.

(3) Both sides of the equation are multiplied (or squared) at the same time, and the equation still holds.

First, the unknown values are substituted into the left and right sides of the equation to calculate their values. Secondly, compare whether the values on both sides are equal and draw a conclusion.

For beginners, it is easy to master the method of solving a linear equation of one variable, but similar to the previous rational number mixed operation, every problem can be done, but it is not guaranteed to be correct.

Therefore, in learning, on the one hand, we should repeatedly pay attention to the law basis of equation deformation and guide the deformation steps with laws, on the other hand, we should constantly pay attention to error-prone points and pursue the simplicity of calculation process.

The application problem of one-dimensional linear equation is mainly to find the equation relationship from practical problems, analyze the known and unknown quantities in practical problems, find out the equation relationship and list the equations, so that students can gradually establish the thinking method of solving practical problems by listing the equations.

Setting method of unknown number:

(1) "Direct demonstration": If an unknown quantity is needed in the topic, it is set as an unknown quantity, which is mostly applicable to the case where only one unknown quantity is needed.

(2) "Indirect setting": For some practical problems, if it is difficult to list the equations by directly setting the unknowns, or the listed equations are complicated, you can choose to indirectly set the unknowns, and the solved indirect unknowns play an intermediary role in determining the required quantities.

(3) "Auxiliary setting": Some application problems not only need to set unknowns directly, but also need to add auxiliary unknowns, but these auxiliary unknowns themselves do not need to be asked, and their function is only to help set equations, and at the same time, in order to find out the real unknowns, they can be eliminated when solving problems.

(4) Conversion between "partial argument" and "overall argument": When the overall argument is difficult, we can consider setting some of them as unknown, and vice versa.

note:

◆ When solving application problems with equations in junior high school, list them as simply as possible (that is, each column of equations directly indicates the meaning of the problem), so that you don't have to worry about too many unknowns, simplify the steps of examining problems and listing equations, and transfer the difficulty to the steps of solving equations.

◆ When setting an unknown number, the unit should be indicated. When listing equations, if the units of data in the problem are not uniform, the units must be converted into uniform units, especially in the trip problem.