Two unknowns need two facts. Each fact becomes one equation.
Find the prices
Three notebooks and two pens cost ₱185. Two notebooks and four pens cost ₱210. How much is one notebook?
Let n be a notebook and p a pen: 3n + 2p = 185 and 2n + 4p = 210. Doubling the first gives 6n + 4p = 370; subtracting the second leaves 4n = 160, so n = ₱40 and p = ₱32.50. Check: 120 + 65 = 185 ✓ and 80 + 130 = 210 ✓ D is the price of a pen — read which item the question asked about.
Two unknowns, two letters, two equations. Never one letter.
let n = cost of a notebook
in pesos
let p = cost of a pen
in pesos
3n + 2p = 185
the first purchase
2n + 4p = 210
the second
Writing down what each letter means, with its unit, takes five seconds and prevents the commonest failure: solving correctly and then answering the wrong question.
Two numbers add to 58
their difference is 12
x + y = 58, x − y = 12
two equations
2x = 70, x = 35
add them
y = 23
35 + 23 = 58 ✓, 35 − 23 = 12 ✓
Sum-and-difference problems always fall out to elimination in one step, because the y terms are already opposites.
A rectangle's length is 4 times its width, and its perimeter is 70 cm. Find the width.
L = 4W, 2L + 2W = 70
"times" gives one equation, the perimeter the other
2(4W) + 2W = 70
substitute L = 4W
10W = 70, W = 7 cm
L = 28 cm; 2(28) + 2(7) = 70 ✓
When one unknown is a multiple of the other, substitute it straight in. A perimeter counts each side twice: two lengths and two widths, not L + W.
30 coins, all ₱5 and ₱10
worth ₱215 in total
f + t = 30
how MANY coins
5f + 10t = 215
how MUCH they are worth
f = 17, t = 13
85 + 130 = 215 ✓
Writing f + t = 215 mixes the two up — that says there are 215 coins. The number of coins and their worth are separate facts, and each gives its own equation.
| Now | In 4 years | |
|---|---|---|
| Ana | a | a + 4 |
| Ben | b | b + 4 |
Ana is 6 years older than Ben
a = b + 6
In 4 years their ages add to 40
(a + 4) + (b + 4) = 40
(b + 6) + b = 32
substitute and simplify
b = 13, a = 19
in 4 years: 23 + 17 = 40 ✓
Check against the words, not the equations you wrote. If you translated a sentence wrongly, the equations will still agree with each other and the answer will still be wrong.
Downstream the speeds add: b + c. Upstream they subtract: b − c.
A banca travels 36 km downstream in 2 hours and returns upstream in 3 hours. Find the speed of the banca in still water and the speed of the current.
downstream: 36 ÷ 2 = 18 km/h
b + c = 18
upstream: 36 ÷ 3 = 12 km/h
b − c = 12
2b = 30, b = 15 km/h
add the equations
c = 3 km/h
15 + 3 = 18 ✓
72 km in 5 hours gives 14.4 km/h, which is neither the banca's speed nor the current's. The two legs have different speeds, so each leg gives its own equation.
Name both unknowns, one sentence per equation — the setups just look different.
A mother is 3 times as old as her daughter. In 12 years she will be twice as old. Find their ages now.
m = 3d
now
m + 12 = 2(d + 12)
in 12 years, both ages grow
3d + 12 = 2d + 24, d = 12
substitute m = 3d
m = 36
Check: in 12 years they are 48 and 24, and 48 = 2 × 24 ✓. Add the 12 years to both people, not just one.
How many litres of 20% juice and 50% juice make 30 litres of 30% juice?
x + y = 30
litres of drink
0.2x + 0.5y = 0.3(30) = 9
litres of pure juice
0.2x + 0.5(30 − x) = 9
substitute y = 30 − x
15 − 0.3x = 9, x = 20, y = 10
Check: 0.2(20) + 0.5(10) = 4 + 5 = 9 ✓. Like coins, one equation counts the amount and the other counts what it is worth.
The digits of a two-digit number add to 9. Reversing the digits makes the number 27 larger. Find the number.
t + u = 9
tens digit t, units digit u
(10u + t) − (10t + u) = 27
the value of each number
9u − 9t = 27, u − t = 3
divide by 9
u = 6, t = 3: the number is 36
add the two equations
Check: 3 + 6 = 9 ✓ and 63 − 36 = 27 ✓. A digit is worth 10 times itself in the tens place.
Two-unknown word problems appear on every periodical test and entrance exam: prices of two items, tickets or coins (count and value), ages, mixtures, and current or wind problems. The fastest route is to name both letters with units, turn each sentence into one equation, then eliminate. For a 10-second check, put the answers back into the words of both sentences, and make sure you answered the quantity that was asked.
Find how many
A cinema sells adult tickets at ₱250 and student tickets at ₱150. It sold 180 tickets for a total of ₱35,000. How many student tickets were sold?
Let a and s be the numbers of tickets: a + s = 180 and 250a + 150s = 35,000. From the first, a = 180 − s. Substituting: 250(180 − s) + 150s = 35,000, so 45,000 − 100s = 35,000, giving 100s = 10,000 and s = 100. Check: 80 adults and 100 students give 20,000 + 15,000 = 35,000 ✓ B is the number of adult tickets.
Hint: one equation counts tickets, the other counts pesos. Write both before solving, and check which one the question asked about at the end.
