Creating Equations (Unknowns in terms of same variable)

In algebra we use variables to represent unknowns.  And, by using the tips and skills in the Creating Equations lesson, we can put together an equation from most word problems.  However, it's difficult (and sometimes impossible) to solve a single equation with two variables.  So, how can we take an equation with two unknowns and write an equation with only one variable?

Often word problems tell you how the two unknowns are related to each other, giving you a way to write one uknown "in terms of" another unknown.

When we say "in terms of" it just means that we are using one variable to define another. Let's try a problem:

Example: Avery is 4 years older than Kielan.  If you add their ages together, they are 16 years old.  How old is Avery?

Let's define our variables:

Avery's age = $a$

Kielan's age = $k$

$a + k = 16$

But, with two variables, we will never know exactly what age either Avery and Kielan are.

BUT!  The problem gives us more information: "Avery is 4 years older than Kielan."  That means that we can give Avery's age in terms of Kielan's age (remember, that just means that we will use Kielan's variable (k) in Avery's variable:

Avery's age $= a = k+4$

Kielan's age $= k$

Now, we can take our original equation and substitute the term $(k-4)$ for the variable $a$ (remember, if things are equal, you can swap them in and out of equations -- they are equal so using a new equal form does not change the equation).

$$\eqalign{a+k&=16\\(k+4)+k&=16}$$

From here, we combine like terms and solve the equation:

$$\eqalign{(k+4)+k&=16\\2k+4&=16\\-4\text{}&\text{}-4\\2k&=12\\\div2\text{}&\text{}\div2\\k&=6}$$

So, Kielan is six years old (and if $a=k+4\text{, then } a=10$) and Avery is 10 years old.

All we did there was write our equation, write one variable in terms of the other, and then substitute our new variable (with the k in it) for the old variable (a).  Then we solve just like we would any algebra equation.

This strategy works for all kinds of problems with two, related unknowns.  Let's try another.

Example: You and a friend rent a car together.  However, because you are the one who wanted to get a fancy car, you pay three times as much as your friend. If the rental car cost \$78, how much did you pay toward the rental?

Let's define our variables:

Your amount = $y$

Your friend's amount = $f$

$y + f = 78$

Again, with two variables, its hard to know who paid how much.

But, the problem gives more information: "..you pay three times as much as your friend."  That means that we can give the amount you paid (y) in terms of the amount your friend paid (f).

Your amount $= y = 3\times f = 3f$

Your friend's amount $= f$

Now, take the original equation and substitute the term $(3f)$ for the variable $y$.

$$\eqalign{y+f&=78\\(3f)+f&=78}$$

From here, combine like terms and solve the equation:

$$\eqalign{(3f)+f&=78\\4f&=78\\\div4\text{}&\text{}\div4\\f&=19.5}$$

$f=19.50$, so, your friend paid \$19.50.  What you paid is equal to $3f$, so you paid: \$58.50.

Overall, when faced with a word problem with two unknowns, see if the unknowns are related.  If they are, you can write one unknown in terms of the other, and then just solve for one variable.  Be careful when writing your final answer: the problem could ask for either the variable or the other term.  So, like this last example, you might have to do some math, using the variable you know, to solve for the other variable.