# 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 =$fy + f = 78$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. #### Practice Problems: • ## Create Equations (Unknowns in terms of the same variable) Write and solve equations for the word problems: 1. You go to a burger joint. A burger costs x. Fries cost .45 less than burgers. If you order 2 burgers and 3 orders of fries, and spend \$17.90, how much does a burger cost? How much do fries cost?
2. At a competing burger joint, drinks cost 50 cents less than fries.  Burgers cost twice as much as fries. If you order 1 burger, one order of fries, and one drink, it costs \$7.70. How much does a burger cost? How much do fries cost? How much do drinks cost? 3. James and Richie took a math test. Richie scored twice as many points as James did. Together, they got 117 points. How many points did Richie score? How many points did James score? 4. James, Richie, and Naseeha all took the same science test. James score 3 times as many points as Richie did. Nasheeha scored 4 more points than James did. All together, they scored 235 points. How many points did James get? How many points did Richie get? How many points did Naseeha get? 5. Bella and Mimi collect trading cards. Mimi has 5 fewer than 3 times as many cards as Bella does. If Mimi has 47 cards, how many cards does Bella have? 6. Khizar and Umar are playing basketball. Umar scores 6 more than 4 times as many points as Khizar does. Together they scored 76 points. How many points does Khizar score? How many points does Umar score? 7. Anthony, Dylan, and Kevin are working on a group project together. Anthony answers half as many questions as Kevin does. Kevin answers 8 fewer questions than Dylan does. If there are 38 questions on the project. How many did Anthony, Kevin, and Dylan each answer? 8. After they finished their project, Anthony, Dylan, and Kevin went out to get lunch. They each bought a sandwich. Dylan's sandwich cost \$1 more than Kevin's sandwich. Anthony's sandwich cost 85 cents less than Kevin's sandwich.  They each got a drink for \$2 each. If they split the cost evenly, and each paid \$7.05, how much did each boy's sandwich cost?