Two unknowns need two equations. The sentences tell you which two — you just have to write them in the right units.
Which system represents this situation?
A hardware store sells extension cords for $14 each and power strips for $9 each. In one week the store sold a total of 120 of these items for a total of $1,380. If c is the number of extension cords sold and p is the number of power strips sold, which system of equations represents this situation?
One equation counts items and one counts dollars. The store sold 120 items altogether, so c + p = 120; the cords brought in 14c dollars and the strips 9p dollars, so 14c + 9p = 1,380. A swaps the two totals, which would mean 1,380 items were sold and they raised $120. B attaches each price to the wrong item — 9c says a cord costs $9. D reads “a total of 120” as a difference; “total” means added, and c − p = 120 would say the store sold 120 more cords than strips.
1
Name the variables in a full sentence before you write anything.
“Let c = the number of extension cords sold.” Not “let c = extension cords”. A letter stands for a number, and which number it stands for decides every equation after it.
a count, not an objectsay the unit out loudone letter per unknown
“c = extension cords” is where these go wrong
If c means the cords themselves rather than how many there are, then 14c reads as “14 cords” instead of “14 dollars per cord, times the number sold”, and the second equation stops meaning anything. The SAT usually defines the letters for you in the stem — read that sentence before you read the question, because it is the only place the units are stated.
2
Shape one: one equation counts the things, the other counts the dollars.
Tickets and money, coins and money, two products and revenue — the same pair of sentences every time.
The count equation
The value equation
the sentence
“a total of 120 of these items”
“for a total of $1,380”
what each term is
how many cords + how many strips
dollars from cords + dollars from strips
the equation
c + p = 120
14c + 9p = 1,380
units on every term
items = items
dollars = dollars
Every term inside one equation has to carry the same unit. Items with items, dollars with dollars, never one of each.
The price never appears in the count equation
c + p = 120 has no 14 and no 9 in it, because 120 is a number of items, not an amount of money. Writing 14c + 9p = 120 is the single most common wrong choice on this item type — and it is the one that looks most like work.
3
Now solve it — then answer the question that was actually asked.
Substitution is quickest here, because the count equation rearranges in one step.
c + p = 120 → p = 120 − c
one unknown in terms of the other
14c + 9(120 − c) = 1,380
put it into the value equation
14c + 1,080 − 9c = 1,380
9 × 120 = 1,080, and 9 × (−c) = −9c
5c = 300, so c = 60
subtract 1,080, then divide by 5
p = 120 − 60 = 60
back into the easy equation
60 extension cords and 60 power strips
Read the last line of the stem again before you pick. The same numbers answer several different questions: p is 60, c − p is 0, and the money taken in from cords alone is 14 × 60 = $840. Underline the quantity being asked for the moment you finish reading, so the finishing step is already decided.
4
Check against the sentence, not against your algebra.
Substituting back into your own equations only proves you solved them correctly. It cannot catch an equation that was wrong to begin with.
“a total of 120 items”60 + 60 = 120the count matches ✓
“for a total of $1,380”14(60) + 9(60) = 840 + 540= 1,380 ✓
does it sound sane?60 and 60, both wholenothing negative, nothing fractional ✓
A negative or fractional answer is an alarm
Half an extension cord, −3 tickets, 7.4 people: the arithmetic can be flawless and the translation still wrong. When the answer cannot exist in the situation, go back to the two sentences rather than back to the algebra.
5
Shape two: two rates, compared.
A gym charges a $60 joining fee plus $25 a month. A second gym charges no joining fee and $35 a month. After how many months do they cost the same?
let m = the number of months, C = the total cost in dollars
name them first, as always
C = 60 + 25m and C = 35m
a one-off fee plus a rate, against a pure rate
60 + 25m = 35m
“cost the same” sets the two expressions equal
60 = 10m, so m = 6
subtract 25m from both sides
C = 35(6) = $210, and 60 + 25(6) = $210 ✓
both plans, same number — the crossing
Left of the crossing the second gym is cheaper; right of it the first one is, because the $60 head start is worth less and less as the $10 a month difference piles up. If the stem asks “for how many months is Plan B cheaper?”, the answer is the side of the crossing, not the crossing itself.
6
Rates that add and subtract: with the wind and against it.
A plane flies 1,200 miles with the wind in 3 hours and the same 1,200 miles against the wind in 4 hours. What is the speed of the wind?
let a = the plane's speed in still air, w = the wind speed, both in mph
two unknowns, so expect two equations
with the wind: 1,200 ÷ 3 = 400 mph
the stem gives distance and time, not speed
against the wind: 1,200 ÷ 4 = 300 mph
same trip, slower because the wind opposes it
a + w = 400 and a − w = 300
the wind helps one way and hinders the other
adding: 2a = 700, so a = 350
the w terms cancel
350 + w = 400, so w = 50 mph
check: 350 − 50 = 300 ✓
A river current, a wind, a moving walkway and an escalator all behave this way: one rate is added on the way out and subtracted on the way back. The give-away is a stem that hands you two times for the same distance — turn each one into a speed before you write a single equation, and the rest is sum-and-difference.
✓
Check: which shape is in front of you?
a count and a valuec + p = 120, 14c + 9p = 1,380items in one, dollars in the other
two rates compared60 + 25m = 35m“the same” sets them equal
helped and hindereda + w = 400, a − w = 300make the speeds first
Remember the flow
Name the variables · One equation per sentence · Match the units · Check the words
Two unknowns always need two equations — and the question rarely wants both of them.
Now you try
Practice problem
After how many minutes?
Two tanks are being filled with water. Tank A already holds 120 gallons and is filling at 8 gallons per minute. Tank B already holds 30 gallons and is filling at 14 gallons per minute. After how many minutes will the two tanks hold the same amount of water?
Let t be the number of minutes. Tank A holds 120 + 8t gallons and tank B holds 30 + 14t, and “the same amount” sets them equal: 120 + 8t = 30 + 14t, so 90 = 6t and t = 15 minutes. Check against the sentence: 120 + 120 = 240 gallons and 30 + 210 = 240 gallons ✓. A is 6, the difference between the two rates — a number you meet on the way, not the answer. C is 25, which comes from adding the two starting amounts instead of subtracting them, 150 ÷ 6. D is 240, the number of gallons each tank ends up holding: the right situation, the wrong quantity.
Hint: write both amounts as “what is already there, plus the rate times the time”, then set the two expressions equal. When you get a number, read the question once more and ask whether it wanted minutes or gallons.
