Now that you know how to set up the equation, the next thing to do is to solve for the value that the question asks for. First and foremost, the most important thing to remember when manipulating equations is to do exactly the same thing to each side of the equation. If you divide one side of an equation by 3, you must divide the other side by 3. If you take the square root of one side of an equation, take the square root of the other.
By treating the two sides of the equation in the same way, you can rest easy that you won’t change the meaning of the equation. You will, of course, change the form of the equation—that’s the point of manipulating it. But the equation will always remain true as long as you always do the same thing to both sides.
For example, let’s look at what happens when you manipulate the equation 3x + 2 = 5, with x = 1.
- Subtract 2 from both sides:
- Multiply both sides by 2:
These examples show that you can tamper with the equation in any way you want, as long as you commit the same tampering on both sides. If you follow this rule, you can manipulate the question how you want without affecting the value of its variables.
Solving an Equation with One Variable
To solve an equation with one variable, 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:
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Solve for x in the equation  + 2 = 4. | |
