On the most basic level, an equation is a statement that one expression is equal to another expression.
When equations contain variables, we can use that statement of equality, along with the basic rules of algebra, to help us determine their value...
When an equation contains just one variable - x, for instance - it’s often a simple process to isolate it - that is, to rearrange the equation such that the variable stands alone on one side of the equals sign. We just need to make sure that the equation remains balanced as we rearrange it.
What does it mean to keep an equation balanced? It means making sure that the original statement of equality given by the equation remains true as we move things around. If we perform an operation - that is:
and we apply that operation to one side of the equation but not the other, we’ve thrown the equation out of balance.
Take this equation as an example.
We can rearrange the equation to solve for x, but we must ensure that the equation stays in balance.
We want to get x by itself, so our first step would be to move the constant 6 from the left side of the equation to the right. We can do this by subtracting it, since subtraction is the inverse of addition.
If we only subtract 6 from one side of the equation, it is no longer in balance; the statement that these two expressions are equal is no longer true. If we subtract 6 from the right side, the equation will be in balance again.
Now, we’ve rearranged our equation while keeping it in balance. All that’s left to isolate the variable - or solve for x - is to divide both sides by 4, since x has a coefficient of 4 in the equation, and the inverse of multiplication is division.
The ability to balance equations is a fundamental skill that you may already be comfortable with, but it’s an important stepping stone for a wide variety of math problems.
To solve more complicated problems, you’ll need to use this skill in conjunction with other math knowledge, such as combining like terms, factoring and distributing, and working with fractions, decimals, exponents, and radicals. As long as you remember why it is important to keep equations in balance, you can tackle any of these problems.
Let’s say you encounter a problem like this one.
The process we need to use to solve for x is the same as before: we must isolate the variable while keeping the equation in balance at every step:
You’ll also see problems with multiple variables. On standardized tests, you may be asked to give the value of one variable “in terms of” one or more other variables.
The variable that you’re asked to express in terms of the other variables is the one you’ll need to isolate.
As always, we just need make sure we keep our equation in balance and we’ll reach the correct answer:
Don’t forget! You have to apply each operation to every term on each side of the equation.
Problems with multiple variables like these are sometimes called “literal” equations, and they can be particularly useful for solving real-world problems. For instance, we can rearrange the equation that gives us the temperature in degrees Fahrenheit in terms of the temperature in degrees Celsius:
Into an equation that gives the temperature in degrees Celsius in terms of the temperature in degrees Fahrenheit:
Rearranging equations like this one can make seemingly complicated mathematical operations easier to comprehend and put to use.