Equations - Things Must be of the Same Type

The following two rules are extremely important, and you may have heard me state them on many occasions.

1. Only things of the same type can be added/subtracted.
2. Only things of the same type can be set equal.

Almost without fail in word problems, units of measure will be attached to various values and variables. In order to appropriately create the equations that are suggested by these word problems, you must be able to attach the appropriate units to each variable/value in the question. Then, keeping in mind the two rules listed above, you should be able to put together the equations correctly. Let's look at the following example and appropriately attach units.

• On a certain day at a furniture store, a total of 149 dressers and tables were sold. If dressers cost $21 each, tables cost $12 each, and a total of $3066 was collected on that day, then how many tables were sold?

Let's establish variables and attach appropriate units.

• Let x = the number of dressers
• Let y = the number of tables
• ^$21⁄_dresser
• ^$12⁄_table
• 149 pieces of furniture
• $3066

Now, let's set up the equations.

• x dressers + y tables = 149 pieces of furniture
• ^$21⁄_dresser*(x dressers) + ^$12⁄_table*(y tables) = $3066

In the first equation above, we are allowed to add the number of dressers to the number of tables because they are both of the same type: pieces of furniture. We can thus set this sum equal to the number of pieces of furniture. In the second equation, by performing the multiplications shown on the left side, we convert each product into something of type $, which is identical to the type on the right hand side of the equation. Using dimensional analysis, we can simplify the second equation to the following.

• ^$21⁄_dresser*(x dressers) + ^$12⁄_table*(y tables) = $3066

We can now drop the struck-through units and simply the equation.

• $21x + $12y = $3066

From here, it is a simple task to solve this via elimination. If you need a refresher on this process, look here.