Setting Up Equations

Setting up equations, or writing equations, involves translating a problem from words into a mathematical statement.

1. Determine what the question is asking.

2. Write down the relevant information in simple statements.

3. Assign symbols to unknown values that need to be found.

4. Determine how the statements relate to each other mathematically.

An exhibit at the zoo contains four times as many giraffes as it has elephants. If the zoo has 12 giraffes, how many animals are there?

1:4 ratio of elephants to giraffes.

1. What is the question asking?

   SHOW ANSWER

   Find the total number of giraffes and elephants.

2. Write down relevant information:

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   Total animals is giraffes and elephants. Four times as many giraffes (More giraffes than elephants).

3. Assign symbols:

   SHOW ANSWER

   Here, we're interested in knowing the total number of animals, and in order to find that, we'll need to know the values for elephants and giraffes, so assign each variable a symbol: A=Animals, E=Elephants, G=Giraffes.

4. While the sentences from step 2 may not win any literary prizes, words like is, and, or times are easy to translate into mathematical symbols ($=, +, \text{ and }\times$, respectively).

   SHOW ANSWER

   There are two equations, one for each sentence. A = G + E, and E * 4 = G.

English and mathematics can be thought of as two separate languages, each with its own symbols, grammar, and style rules. Setting up an equation is similar to translating a paragraph between two languages. The result should contain the same information as the original piece. A direct translation is not always possible, because words that exist in one language may not exist in another, or a word-for-word translation may not make sense in the final language.

Write two equations from the following information:

A bat and ball cost $100, but the bat costs $90 more than the ball.

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Bat + Ball = 100
Bat - Ball = 90. $_\square$

In the above equations, we have two unknowns: the price of the bat and the price of the ball. The unknown parts of the equation, or variables, may have one or more answers, depending on the problem. Variables are usually symbolized with a letter in algebraic equations.

Rewrite the two equations using $x$ and $y$ to represent the variables.

Bat + Ball = 100
Bat - Ball = 90

$x=$ Bat
$y=$ Ball

$x+y=100$
$x-y=90$. $_\square$

Story problems can contain lots of information, including details that are not necessary to solve the problem. Additionally, the problem may assume general knowledge on the reader's part, and not explain all the information explicitly. Writing that information down in a streamlined form is an efficient way to solve problems.

At the beginning of January, Johnny and Mary decide they want to buy a new car together. The car costs $8,000 and they want to buy it in July. Each of them will pay half the total amount. Mary has $2,000 in savings and her income is $1,200 per month. Johnny has no savings and his income is $2,500 per month. How much money should Johnny save per month in order to reach this goal?

In reading this question, the person solving it may start streamlining information in her head. January to July is six months. They need a total of $8,000 and have $2,000, so they need another $6,000, or $1,000 per month. While this answer is true, it's not correct, because the solver tackled the wrong question. The problem asks how much money Johnny needs to save per month.

Another reader might be overwhelmed by the number of details in the problem, and not start with an equation in their head at all. This reader might try to guess how much money Johnny would need per month. Let's see, if Johnny saves $1,000 per month for six months, he would have $6,000 by the time they go to buy the car. That's more than enough. While that solution may work in real life, it may also leave Johnny without enough money left to pay his rent.

Neither of these students is going to get points on an exam, or give Johnny the answer he's really looking for.

While no single detail in this problem is complicated, the number of facts the student is being asked to remember may keep her from finding the answer quickly and efficiently without writing out an equation. Additionally, the details presented about Mary are distracting.

Translating words into symbols is a great way to get started solving word problems. The above example asks us to find the money Johnny needs to save. The problem states that he needs to have half of the $8,000 by July. Those two sentences can be rewritten as follows:

$$Johnny = J = (\$8,000/2) = \$4,000 \implies J = \$4,000.$$

Johnny needs $4,000 by July. He has six months to save, and needs to save an unknown number of dollars per month. Assigning the unknown amount the symbol $x$ allows those pieces of information to be added to the equation. The result is a single equation with a single variable, which can be solved with division.

$$\begin{aligned} \$4,000 &= 6x\\ \$4,000/6 &= 6x/6\\ \Rightarrow \$667 &= x. \end{aligned}$$

As the problems become more complicated, multiple steps or multiple variables may be involved. A problem may also require writing and solving several equations, using information found in one step to solve the next.

Now try solving some equations once you get them set up!