Interpret Equations

Equations are mathematical sentences.  We write equations to solve for variables that we don't know, but can predict based upon other variables. Some of the most useful -- and difficult -- math problems ask students to write or interpret equations.  What does this variable mean?  What happens to variable $n$ when variable $x$ goes up or down?

Equations based on real life (or on word problems that simulate real life) are number sentences that represent real situations. One of the best way to interpret these equations is to break the math sentence apart and turn it back into the real life situation it represents.  From there, it's often easy to figure out what variables represent what, and what happens to one variable when another increases or decreases. 

Sometimes the best way to solve this kind of problem is to think logically about how you would set up an equation (not using variables, just using the ideas from the question) and then translating that into variables.

Let's try an example:

Kaitlin has a summer job writing reading curriculum based on the new reading standards. The number of standards she still has to write curriculum for can be estimated with the equation $S=107-3x$, where $S$ represents the number of standards she still needs to write curriculum for and $x$ represents the number of days she has worked so far this summer.  

What is the meaning of 3 in this equation?

What is the meaning of 107 in this equation?

What happens to $S$ when $x$ increases?

To interpret this equation, let's try to figure out what's going on. Kaitlin is working on some number of standards. $S$ represents the number of standards she has left. We must assume that $S$ will go down as the summer proceeds. So, our basic equation would be:

$\boxed{\text{Standards left}}=\boxed{\text{Standards started with}}-\boxed{\text{Standards worked on}}$

That's just the logic of the problem. How does it match up with the equation that the problem provided?

$S=107-3x$

We know that $S$ represents "Standards left" so, so far, our equations match up.

 $S=\boxed{\text{Standards started with}}-\boxed{\text{Standards worked on}}$

Now, how can we relate "Standards started with" and "Standards worked on" with $107-3x$?  We know that $x$ represents the number of days that Kaitlin has worked. Each day she works, she finishes some standards. How many standards does she finish each day?  Let's say that she finished 2 standards a day, and she worked for 3 days, how many standards would she finish? 6! Right.  How did you know that?  You multiplied them together. So, what do you think $3x$ represents?

If $x$ is number of days and it's multiplied by 3, Kaitlin must accomplish 3 standards a day.

$S=\boxed{\text{Standards started with}}-3x$

From there, you can see what 107 must represent: the number of standards that Kaitlin started with. From here, it's easy to answer the first two questions:

What is the meaning of 3 in this equation?

3 is the number of standards the Kaitlin completes each day she works. 

What is the meaning of 107 in this equation?

107 is the total number of standards that Kaitlin has to write curriculum for.

What about the last question?

What happens to $S$ when $x$ increases?

Let's think about it. $S$ is the number of standards that Kaitlin still needs to work on. $x$ is the number of days that she has worked. As she works more days, what will happen to the number of standards that she has to work on?  It will decrease. 

In the first example, it's helpful to try to think, logically, about how you would try to solve the problem. Other times, it's easier to just translate the equation that is given, using the variable definitions that are provided in the question. 

Let's try one more example:

A restaurant kitchen has to figure out prices for birthday parties. They use the expression $g(m+d)+250h$ to calculate the total price for a party. $g$ represents the number of guests who will attend the party, $h$ represents the number of hours that the party will last, $m$ represents the average cost of the main dishes that will be offered to guests, and $d$ represents the average cost of the desserts that will be offered to the guests.

Which of the following is the best interpretation for the meaning of the 250 in the equation?

How would you rewrite the equation if the party hosts decided not to offer dessert?

Which variable will change the most if the hosts decide to offer their guests the 3 most expensive entrees on the menu?

Let's start with the given equation:

$g(m+d)+250h$

Now, let's translate that into words:

$\boxed{\text{guests}}(\boxed{\text{main dish cost}}+\boxed{\text{dessert cost}})+250(\boxed{\text{hours of party}})$

So, what's going on here? If we look at the parentheses, we see that the first thing that we we do is add the cost of the main dish and the dessert. Then we multiply that by the number of guests. That makes sense: find out how much people's meals should cost, then multiply that by the number of people.

Then to that total, you add 250 times the hours of the party. What's going on there? The restaurant must charge 250 for each hour of party. So, a 1 hour party adds 250 to the cost of the food. A 2 hour party adds 500 to the cost of the food, etc. 

Ok, so we understand the equation, let's answer the questions:

Which of the following is the best interpretation for the meaning of the 250 in the equation?

We already answered this: 250 is the amount the restaurant charges per hour of the party. 

How would you rewrite the equation if the party hosts decided not to offer dessert?

No dessert, then you don't have to pay for dessert. Which variable represents dessert? $d$.  Take that out.  Or make it a zero.

Which variable will change the most if the hosts decide to offer their guests the 3 most expensive entrees on the menu?

What variable captures the price of the entrees? That's the main dish. So that's captured by $m$, which is the average of the cost of the entrees they will offer. If they offer very expensive entrees, then $m$ will be higher. 

Many students find these types of word problems intimidating. Don't be afraid of them. Write them out. Label them and translate variables into things. Then, think logically: what would be added to what? What would be multiplied to what? Substitute some real numbers if that helps you.