➕ Algebra 1 · Systems of Equations
Systems Word Problems: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Matches versus points
In many soccer leagues, a win earns a team 3 points, a draw earns 1 point and a loss earns 0 points. Suppose a team has played 12 matches without a loss, so every match was a win or a draw, and it has 30 points. Let w be the number of wins and d be the number of draws. Matches: w + d = 12. Points: 3w + d = 30. Subtract the first equation from the second: 2w = 18, so w = 9 and d = 3. Check: 27 + 3 = 30.
How it connects: The first equation counts matches and the second counts points, so each equation keeps one unit: 12 matches is not 12 points. The limit: the points for a win and a draw depend on the league and the sport (many leagues use 3 and 1, but a problem has to tell you the values). And this team has no losses. If losses were allowed, there would be three unknown counts, and you would need another fact.
Apples and oranges
People say “that is comparing apples and oranges” when two things are too different to compare directly.
How it connects: An equation must compare like with like. You cannot set tickets equal to dollars, so s + a = 900 is wrong when 900 is a number of dollars. The limit: you can add apples and oranges if you count the same unit for both, such as pieces of fruit. That is why s + a = 120 is fine: student tickets and adult tickets are both tickets. And 6s and 10a are both dollars, so they can be added too.
Look before you leap
This proverb advises you to think carefully about what you are doing before you act.
How it connects: Define your variables and write your equations before you start solving. The limit: the proverb does not say what to look for. In a word problem, look for the unknowns, the unit of each quantity, and the two facts in the story.
Guess my two numbers
In a guessing game, your friend says, “My two numbers add up to 21, and the larger minus the smaller is 7.” The first clue alone fits many pairs: 20 and 1, 15 and 6, 10 and 11, and more. The second clue narrows it to one pair: 14 and 7.
How it connects: Two unknowns need two clues. Let x be the larger number and y the smaller: x + y = 21 and x − y = 7. Adding the equations gives 2x = 28, so x = 14, and then y = 7. The limit: two clues do not always pin down one pair. A clue that repeats the first one adds nothing (infinitely many pairs), and a clue that contradicts it, like “they add to 25,” leaves no pair at all.
Punch card or pay per cup
A smoothie shop offers a punch card for 12 dollars once. With the card, each smoothie costs 3 dollars. Without the card, each smoothie costs 5 dollars. For n smoothies, the card plan costs 12 + 3n dollars and the no-card plan costs 5n dollars. They are equal when 12 + 3n = 5n, so n = 6, and both cost 30 dollars. With fewer than 6 smoothies, skipping the card is cheaper. With more than 6, the card wins.
How it connects: The crossing point is where the two cost formulas are equal. Before it one plan is cheaper, after it the other is. The limit: smoothies come in whole numbers, and the story assumes the prices stay steady. If the two plans charged the same price per smoothie but different fees, the costs would never be equal.
The half ticket
A system says a theater sold 7.5 tickets. The math is perfect. But who bought the half ticket? Someone who only wanted to see the first half of the show.
How it connects: The algebra can be correct and the answer still not make sense. A count of tickets must be a whole number, so 7.5 tickets means the equations or the numbers in the problem are wrong, or the situation is impossible. The limit: some quantities can be fractions, like hours or kilometers. Always ask what the variable counts.
Memory help
SOLVED
Say in words what each variable means (with units). One equation for each fact in the story. Look for the easiest method: substitution when a variable is alone, elimination for standard form, a graph for the picture. Values: solve for both variables. Examine the answer against the original story. Deliver the answer in a sentence with units.
The count-and-value table
Draw a table with a column for each kind of thing and a column for the total. Row 1 is how many (the count). Row 2 is what each one is worth. Row 3 is the value: price times number. Any row that has a total gives you one equation. Write the unit in each row so you never mix tickets with dollars.
Name it, write two
Name the unknowns, write two facts; keep the units straight, no cracks. Solve for both, then check the tale; say it in a sentence, and set sail.
A count is not a value
12 matches is not 12 points. In s + a = 120 the unit is tickets. In 6s + 10a = 900 the unit is dollars. If a number in your story is a count, it goes in the count equation. If it is a total of money, distance or points, it goes in the value equation.
Do not round a nonviable answer
If the equations give 7.5 tickets, rounding to 8 breaks the totals. Recheck the equations and the numbers in the problem. If they are right, the story as written is impossible.