you must isolate that variable. Isolating a variable means manipulating the equation until the variable is the only thing remaining on one side of the equation. Then, by definition, that variable is equal to everything on the other side, and you have successfully “solved for the variable.” For the quickest results, take the equation apart in the reverse order of operations. That is, first add and subtract any extra terms on the same side as the variable. Then, multiply and divide anything on the same side of the variable. Next, raise both sides of the equation to a power or take their roots according to any exponent attached to the variable. And finally, do anything inside parentheses. This process is PEMDAS in reverse (SADMEP!). The idea is to “undo” everything that is being done to the variable so that it will be isolated in the end. Let’s look at an example:
In this equation, the variable x is being squared, multiplied by 3, added to 5, etc. We need to do the opposite of all these operations in order to isolate x and thus solve the equation. First, subtract 1 from both sides of the equation:
Then, multiply both sides of the equation by 4:
Next, divide both sides of the equation by 3:
Now, subtract 5 from both sides of the equation:
Again, divide both sides of the equation by 3:
Finally, take the square root of each side of the equation:
We have isolated x to show that x = ±5. Sometimes the variable that needs to be isolated is not conveniently located. For example, it might be in a denominator or an exponent. Equations like these are solved the same way as any other equation, except that you may need different techniques to isolate the variable. Let’s look at a couple of examples:
Solve for x in the equation + 2 = 4.
The key step is to multiply both sides by x to extract the variable from the denominator. It is not at all uncommon to have to move the variable from side to side in order to isolate it. Remember, performing an operation on a variable is mathematically no different than performing that operation on a constant or any other quantity. Here’s another, slightly more complicated example:
This question is a good example of how it’s not always simple to isolate a variable. (Don’t worry about the logarithm in this problem—we’ll review these later on in the chapter.) However, as you can see, even the thorniest problems can be solved systematically—as long as you have the right tools. In the next section, we’ll discuss factoring and distributing, two techniques that were used in this example. So, having just given you a very basic introduction to solving equations, we’ll reemphasize two things:Do the same thing to both sides. Work backward (with respect to the order of operations). Now we get into some more interesting tools you will need to solve certain equations. Distributing and Factoring Distributing and factoring are two of the most important techniques in algebra. They give you ways of manipulating expressions without changing the expression’s value. So it follows that you can factor or distribute one side of the equation without doing the same for the other side of the equation. The basis for both techniques is the following property, called the distributive property:英语作文
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