Let’s say that you have the equation and you are asked to “solve for x”. Before you do anything – ask yourself “what does it mean to solve an equation anyway?”.

Second, just to get the language of math down, this is a linear equation since the variable x has an exponent of one, and the entire equation just involves x and a constant (a number). A linear equation is likely the most simple type of equation you will encounter and comes up a lot in basically every level of math! Some examples of linear equations are:

In each example above – even when there were many terms, no variable has an exponent. For all linear equations, no matter how simple or complicated looking, the idea is to isolate the variable. You want to use the basic rules of algebra to get something like:

variable = number

There many be many operations to get there, but that is the end goal. Essentially you will have to first simplify by combining like terms, or using properties like the distributive property and then undo whatever operations involve .

Example Solve for :

If you just think about this, you could probably figure out the answer without doing any calculations on paper. You want to find a number that is when you mulitply it by . Obviously this will be since . But you should know the process to get this answer so that you can solve the more complicated problems.

Looking at the equation , both sides are simplified. Therefore, the next step is to isolate by undoing whatever operations involve . Right now, is being multiplied by , therefore you will DIVIDE by (this is the same as multiplying by the reciprocal of ). This is OK as long as you remember an important rule: Whatever you do to one side of the equation MUST be done to the other side of the equation.

The plan is to divide by on both sides:

(since 12/4=3).

Example Solve for :

Solution:

(since 9 is being subtracted from x, you add 9 to both sides to undo it)

.

If there is more than one operation, it is important to remember the order of operations, PEMDAS. Since you are undoing the operations to x, you will work from the “outside in”, in other words the opposite order of the order of operations.

Example Solve for :

Solution:

(add 7 to both sides)

(divide by 2 on both sides)

.

More Advanced Examples

In the following examples, there are more variable terms and possibly some simplification that needs to take place. In each case, the steps will be to first simplify both sides, then get all the variables on one side of the equation, and finally follow the steps above.

Example Solve for :

Since both sides are simplified, the next step is to get all of the x’s on one side of the equation and all the numbers on the other side. The same rule applies – whatever you do to one side of the equation, you must do to the other side as well! It is possible to either move the or the . Suppose you moved the . Since it is positive, you would do this by subtracting it from both sides:

Now the equation looks like those that were worked before. The next step is to subtract 2 from both sides:

Finally, since , divide both sides by :

Example Solve for : .

Solution:

(simplify both sides)

(since 2x is negative, add it to both sides)

(subtract 5 from both sides)

(divide by 5 on both sides)

.

Many Solutions or No Solutions

There are times when you follow all of these steps and a really strange solution comes up. For example, when solving the equation following the steps above you will end up with . This is certainly true but what good does it do?

If you get a statement such as this, it means that the equation has infinitely many solutions. Any x you could think of would satisfy the equation . The appropriate answer in this case is “infinitely many solutions”.

The other situation comes up when you simplify an equation down into a statement that is never true such as or. This happens with the equation which will lead to , something that is certainly never true. This means that no x would satisfy this equation. In other words “no solution”.

 

Related Posts:

• Sketching Quick and Accurate Graphs of Linear Equations

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