Translate real-world scenarios into algebraic expressions by identifying unknown quantities and the operations that relate them. Focus on precision in choosing variables and defining what they represent.
A lesson of Algebra 1, in Foundations of Algebra, about 35 minutes.
By the end of this lesson, you will be able to take a situation described in words, define a variable precisely, express every other quantity in terms of it, and write the expression or equation that models the situation.
Setting It Up Is the Hard Part
In most word problems, the algebra at the end is the easy bit. What stops
people is the gap between the paragraph and the first line of symbols.
Turning a situation into algebra is called modeling, and it is the reason
mathematics is useful outside a classroom at all. Engineers, doctors and
economists all do the same thing: read a messy situation, decide what the
unknown is, and write down a relationship that captures it.
There is a reliable habit that makes it work, and it is smaller than it sounds.
Read carefully. Define the variable precisely. Write the algebra to match
the words exactly.
Four Steps, in This Order
Do all four before writing a single symbol.
Step
What you do
Why it matters
1. Identify
Find what is unknown
Says what you are solving for
2. Name
Write "Let $x$ be ..." with units
Removes every ambiguity later
3. Express
Write the other quantities in terms of that letter
Links the unknowns together
4. Write
Build the expression or equation
Captures the relationship
Step 2 is the one people skip, and it is the one that decides whether the rest
works.
Naming the Variable Properly
A definition is strong when nobody, including you an hour later, could read it
two ways.
Weak
Strong
Let $x$ be money
Let $x$ be the amount Anya earns per hour, in dollars
Let $n$ be numbers
Let $n$ be the smaller of the two consecutive integers
Let $t$ be time
Let $t$ be the number of hours the car has been driving
A strong definition says what the quantity is, in what units, and
when, if the timing matters. It costs one line, and it is what makes your
final answer interpretable as something in the real world rather than just a
number.
Example 1: Two Unknowns, One Letter
Here is the whole method on one problem, worked all the way through.
Marcus scored 8 more points than Deja. Together they scored 54.
Both scores are unknown, but they are not independent. Once you know Deja's,
you know Marcus's, so one letter is enough.
Let $d$ be Deja's score, in points.
Write Marcus's score in terms of d d + 8
Write the total of the two scores d + (d + 8)
Say that the total is 54 d + (d + 8) = 54
Worked Example 1: The Same Shape, Your Turn
Priya read 12 more pages than Sam. Between them they read 86 pages.
Let $s$ be the number of pages Sam read.
Example 2: Consecutive Integers
Consecutive integers go up by one each time, so once you name the smallest, the
others are forced.
The sum of three consecutive integers is 78.
Let $n$ be the smallest of the three.
Write the second integer n + 1
Write the third integer n + 2
Write the sum of all three, set equal to 78 n + (n + 1) + (n + 2) = 78
Simplify the left hand side 3n + 3 = 78
Worked Example 2: Consecutive Even Numbers
Even numbers are two apart, not one, and that is the only change.
The sum of three consecutive even integers is 96.
Let $n$ be the smallest of the three.
Example 3: A Fixed Charge Plus a Rate
Almost every pricing problem has the same two parts: something you pay once,
and something you pay for each unit.
A plumber charges $$75$ to turn up, plus $$45$ for every hour of work.
Let $h$ be the number of hours worked.
Write the part of the bill that depends on the hours 45h
Add the part that is the same whatever happens 45h + 75
Write the equation saying the bill came to 210 dollars 45h + 75 = 210
Worked Example 3: A Taxi Fare
A taxi charges $$2.50$ to get in, then $$1.75$ per mile. A ride cost $$16.50$.
Let $m$ be the number of miles traveled.
Worked Example 4: A Shape with a Rule in It
A rectangle's length is 5 more than twice its width. Its perimeter is 52 cm.
Let $w$ be the width, in centimeters.
Example 4: One Problem, All Four Steps, Start to Finish
Here is a full solution written out the way you would write it on paper.
A phone plan costs $$20$ per month plus $$0.10$ for each text. Rosa's bill
was $$27.50$. How many texts did she send?
Identify. The number of texts is unknown.
Name. Let $t$ be the number of texts sent in the month.
Express. The texts cost $0.10t$ dollars. The monthly fee is $20$ dollars,
and it does not depend on $t$.
Write. The bill is the two parts added together:
$$0.10t + 20 = 27.50$$
Solve.
$$0.10t = 7.50$$
$$t = 75$$
Check against the words, not just the equation. Seventy five texts at ten
cents each is $$7.50$, plus the $$20$ fee, which is $$27.50$. That is what
the problem said the bill was, so the answer stands.
That last step is worth the ten seconds. An answer can satisfy your equation
perfectly and still be the wrong quantity, if the equation was set up wrong.
Common Mistakes
Mistake 1: A vague variable definition
Wrong: Let $x$ be the age.
Right: Let $x$ be Maya's current age, in years.
Whose age, and when? Without that, the expression you build next is a guess.
Mistake 2: One letter doing two jobs
Maya is 5 years older than Zara.
If $x$ is Maya's age, Zara's is $x - 5$. If $x$ is Zara's age, Maya's is
$x + 5$. Both are correct, and they are not the same equation, so you have to
say which one you chose.
Mistake 3: Reversing a "less than" phrase
Wrong: "8 less than a number" written as $8 - x$
Right: "8 less than a number" is $x - 8$
You are taking $8$ away from the number, not taking the number away from $8$.
Mistake 4: Leaving a part of the total out
A taxi charges $$2.50$ plus $$1.75$ per mile.
Wrong: $1.75m$
Right: $1.75m + 2.50$
The fixed charge is easy to lose because it is not attached to a letter. Read
the sentence to the end before you write anything.
Mistake 5: Answering the equation instead of the question
If you set $w$ to be the width and the question asks for the length, then
finding $w = 7$ is not the answer. Go back to the words and check which
quantity was asked for.
Where This Goes Next
A single letter standing for one unknown is the first job variables do. There
are two more coming.
Parameters. In $y = mx + b$, the letters $x$ and $y$ vary along a line
while $m$ and $b$ stay fixed for that particular line. Same alphabet, different
role, and the role is decided by what is being held constant.
Systems. When two unknowns are genuinely independent, one letter is not
enough. Two letters and two equations describe the situation, and solving them
together is the next stage of this same skill.
The habit is what carries forward. Students who write "let $x$ be ..." with a
full description almost never lose track of what their answer means. Students
who skip it usually do, and usually near the end where it costs the most.
Check Your Understanding
Five of them, no instructions this time.
Summary
The four steps:
Identify what is unknown
Name it: "Let $x$ be ...", with units
Express every other quantity in terms of that letter
Write the expression or equation
The setups worth recognizing on sight:
Situation
Name this
Then write
Two related quantities
The smaller one
The other one in terms of it
Consecutive integers
The smallest, $n$
$n+1$, $n+2$, or $n+2$, $n+4$ if even or odd
A fee plus a rate
The changing quantity
fixed fee $+$ rate $\times$ quantity
A shape with a rule
The simpler side
The other side in terms of it
At the end: check your answer against the original words, and make sure the
quantity you found is the one that was asked for.