What is empathy?
Example 1? Solve the equation 5x+2=7x-8.
In order to make the equation appear in the form of ax=b, we must combine similar terms, but they are not on the same side of the equal sign. How to merge them? Let's take advantage of the basic properties of the equation, subtract 2 from both sides of the equation, and then subtract 7x from both sides of the equation to get: 5x-7x=-8-2, and then merge similar terms.
Here, 7x changes the sign and moves it to the left of the equation, and 2 changes the sign and moves it to the right of the equation. This deformation is equivalent to changing the sign of an item in the equation and moving it from one side of the equation to the other. This deformation is called the moving term.
Second, how to move items
Let's look at the above example first: solve the equation 5x+2=7x-8.
Analysis: In order to make the form of the equation ax=b, we can move the unknown term to the left of the equation, the known term to the right of the equation, or move the unknown term to the right of the equation and the known term to the left of the equation, so we can get the following two solutions according to the law of shifting terms.
Solution 1: Move the term to get 5x-7x=-8-2, merge the similar terms to get -2x=- 10, and the coefficient reaches 1 to get x=5.
Solution 2: Shift the terms to get 2+8=7x-5x, combine similar terms to get 10=2x, and the coefficient is 1, and get x=5. (Finally, check the roots by oral calculation)
Combining solution 1 solution 2, we can sum up what is the law of shifting terms when solving a linear equation with one variable like this. (Generally, items with unknowns are moved to one side, and items without unknowns are moved to the other side). It is customary to move items containing unknowns to the left, and sometimes to the right for simplicity.
Comparing the two solutions, the unknown term moves in different directions, but both can simplify the equation to the simplest form ax=b, and then find the solution of the equation.
Transposition-related interpretation
Why do you have to change the sign of the moving item to move from one side of the equation to the other?
We can understand it this way:
According to the law of subtraction: a-b=a+(-b), that is, subtracting a number is equal to adding the reciprocal of this number.
When we want to move an item (such as x) on the left to the right, we actually subtract the item (x) on the left. According to the same solution principle, we must also subtract this term on the right, and then according to the law of subtraction, the opposite number of this term (x) must be added to the right. Therefore, after the item (x) on the left is subtracted (from nothing), its opposite number will appear on the right (. I feel that the item on the left has changed its number and moved to the right. It is the same reason to move some terms on the right side of the equation to the left.