Systems of Equations in Word Problems
A word problem leads to a system of equations when it has two unknown quantities and two separate facts connecting them. In brief: if 3 notebooks and 2 pens cost 13 dollars and 1 notebook and 4 pens cost 11 dollars, define n and p for the prices and solve 3n + 2p = 13 and n + 4p = 11. The solution is n = 3 and p = 2.
The hardest part is not the algebra but the translation from words to equations. A reliable routine makes it easy: define the unknowns, write one equation per fact, solve and check against the original text. This guide shows the routine and works through typical problem types: prices, mixtures, ages, motion and numbers, followed by a table of the usual equation patterns.
How do you solve a word problem with a system?
- Read the problem and underline what is asked.
- Define a letter for each unknown, with units, such as “n = price of a notebook in dollars”.
- Write one equation for each fact in the text.
- Solve the system with substitution or elimination.
- Check the solution in the words of the problem, not only in the equations.
- Answer with a sentence and units.
How do you solve price and quantity problems?
Example 1: notebooks and pens
3 notebooks and 2 pens cost 13 dollars. 1 notebook and 4 pens cost 11 dollars. Let n and p be the prices. Equations: 3n + 2p = 13 and n + 4p = 11. From the second, n = 11 − 4p. Substitute: 3(11 − 4p) + 2p = 13, so 33 − 12p + 2p = 13, giving −10p = −20 and p = 2. Then n = 11 − 8 = 3. Check: 3 · 3 + 2 · 2 = 13 ✓ and 3 + 8 = 11 ✓. A notebook costs 3 dollars and a pen 2 dollars.
Example 2: tickets
A cinema sold 120 tickets for 1,020 dollars. Adult tickets cost 10 dollars and child tickets cost 6 dollars. Let a be adults and c children. Then a + c = 120 and 10a + 6c = 1020. From the first, c = 120 − a. Substitute: 10a + 720 − 6a = 1020, so 4a = 300 and a = 75, c = 45. Check: 750 + 270 = 1020 ✓.
How do you solve mixture problems?
For mixtures write one equation for the total amount and one for the amount of the active substance.
Example 3: solutions
How much 10% and 40% salt solution are needed to make 30 litres of 30% solution? Let x and y be the litres of each. Then x + y = 30 and 0.10x + 0.40y = 0.30 · 30 = 9. From the first, x = 30 − y. Substitute: 3 − 0.1y + 0.4y = 9, so 0.3y = 6 and y = 20, x = 10. Check: 1 + 8 = 9 ✓. Use 10 litres of the 10% solution and 20 litres of the 40% solution.
How do you solve motion and age problems?
Example 4: boat in a river
A boat travels 24 km downstream in 2 hours and returns upstream in 3 hours. Let b be the boat’s speed in still water and c the speed of the current, both in km/h. Then b + c = 24/2 = 12 and b − c = 24/3 = 8. Adding: 2b = 20, so b = 10 and c = 2. Check: 12 and 8 ✓.
Example 5: ages
A mother is 4 times as old as her son. In 6 years she will be 3 times as old as he will be. Let m and s be the present ages: m = 4s and m + 6 = 3(s + 6). Substituting: 4s + 6 = 3s + 18, so s = 12 and m = 48. Check: in 6 years, 54 and 18, and 54 = 3 · 18 ✓.
Example 6: digits
The sum of the digits of a two-digit number is 11. Swapping the digits gives a number that is 27 smaller. Let t and u be the tens digit and the units digit: t + u = 11 and (10u + t) = (10t + u) − 27, which gives 9t − 9u = 27, so t − u = 3. Adding: 2t = 14, so t = 7 and u = 4. The number is 74, and swapped it is 47 = 74 − 27 ✓.
Typical equation patterns
| Type | First equation | Second equation |
|---|---|---|
| Prices | count of items | price × count = total |
| Mixtures | total amount | amount of substance |
| Motion with current | b + c = downstream speed | b − c = upstream speed |
| Ages | relation now | relation after some years |
| Digits | sum of digits | value of number |
Where do such problems come from?
Word problems that give two conditions about two quantities are very old. Babylonian tablets and Chinese texts such as The Nine Chapters on the Mathematical Art contain problems about grain, animals and prices solved by methods equivalent to a system of equations. Diophantus of Alexandria collected many such problems in his Arithmetica, though his main interest was in integer or rational solutions.
Where are they used today?
Budgeting, pricing and planning production use the same translation from facts to equations. In chemistry, mixtures and dilutions are solved this way, and in logistics a system can balance the numbers of two kinds of vehicles against load and cost limits. Larger real problems use many unknowns and software, but the thinking is the same.
Which mistakes are common?
Defining unknowns vaguely
Write “x = number of adult tickets” and not “x = adults”. The exact definition prevents mixing up the equations.
Mixing units
Convert minutes to hours and percentages to decimals before writing equations.
Writing two equations from the same fact
Each equation must come from a different piece of information. Otherwise the system is dependent.
Not checking in the text
A negative number of tickets or a fractional number of people signals an error. Check that the answer makes sense.