HighSchoolQ

➕ Algebra 1 · Systems of Equations

Systems Word Problems: notes and common mistakes

Study notes

Key ideas

  • A system word problem has two unknown quantities and two facts. Each fact becomes one equation.
  • Define each variable in words, with units, before you write equations: “let s be the number of student tickets sold.”
  • A count equation adds how many. A value equation adds what each kind is worth (price or rate times number). Keep one unit inside each equation.
  • Choose a method: substitution when a variable is alone, elimination for standard form, graphing for the picture. Every correct method gives the same solution.
  • Check in the original words, not only in your equations: whole numbers for counts, no negative amounts, and the question that was asked.
  • State the answer in a sentence with units. If the solution is not sensible, first recheck your equations and the numbers in the problem. If they are right, the situation as described is impossible.

Steps

  1. Define each variable in words, with its unit.
  2. Write one equation for each fact: a count equation, a value equation, a plan equation or a distance equation.
  3. Choose a method and solve for both variables.
  4. Check in the original words: does it fit every sentence, and is it sensible (whole, non-negative counts)?
  5. Answer in a sentence with units.

Formulas

  • Count equation: (number of A) + (number of B) = total number
  • Value equation: (price of A) × (number of A) + (price of B) × (number of B) = total value
  • Plan cost: cost = fixed fee + (rate) × (amount)
  • Distance: distance = rate × time

Watch out

  • Do not mix units in one equation: do not add a count of tickets to a total in dollars.
  • A count is not a value: 12 matches is not 12 points.
  • Do not round a nonviable answer (like 7.5 tickets) to make it fit. The rounded numbers break the totals.
  • Answer the question that was asked, and solve for both variables so you can check.
  • Checking only in your own equations misses a mistranslation. Reread the story.

Why it works

  • Why two equations? One equation with two unknowns has many solutions. A second fact narrows it to one pair, except in the special cases of no solution or infinitely many.
  • Why check in the story? Your equations are your translation of the story. If the translation is wrong, the equations will agree with a wrong answer.
  • Why can a correct solution of the equations be wrong for the story? The equations do not know that tickets are whole numbers. The story adds that extra rule.

Common mistakes

✗ Mistake“Let s = tickets.”
✓ InsteadSay what the letter counts or measures, with its unit: “let s be the number of student tickets sold.”
✗ MistakeA shop sold 25 sandwiches and wraps for 210 dollars, so s + w = 210.
✓ InsteadThe 25 is a count and the 210 is dollars. The count equation is s + w = 25, and the dollar equation needs the prices.
✗ MistakeThe equations give 7.5 tickets, so round up to 8.
✓ InsteadRounding breaks the totals. Recheck the equations and the numbers in the problem. If they are right, the situation as described is impossible.
✗ MistakeMy answer works in both of my equations, so it must be right.
✓ InsteadThat only checks the algebra. Reread the story to make sure each equation matches a sentence, and check that the answer is sensible.
✗ MistakeTwo plans with different fees always cost the same at some point.
✓ InsteadIf the rates are equal and the fees differ, the costs are never equal. If the crossing falls at a negative number of months, they are never equal in the story. If it falls between whole months and you pay by whole months, the costs are never exactly equal, but the cheaper plan still switches.
✗ MistakeI found x = 3, so I am done.
✓ InsteadA system has two unknowns. Find both, check them in the story, and answer in a sentence that says what each number counts.

Teach it back

Explain to a friend how to solve this: “A cafe sold 22 lunches in one day. Salads cost 8 dollars each and sandwiches cost 5 dollars each, and the sales were 137 dollars.” Then say how you know the answer makes sense.

Explaining it out loud shows what you really know. Did you…

  • I defined s as the number of salads sold and w as the number of sandwiches sold.
  • I wrote a count equation, s + w = 22, and a dollar equation, 8s + 5w = 137.
  • I substituted w = 22 − s to get 3s + 110 = 137, so s = 9 and w = 13.
  • I checked in the story: 9 + 13 = 22 lunches and 8(9) + 5(13) = 72 + 65 = 137 dollars, and both counts are whole numbers.
  • I answered in a sentence with units: the cafe sold 9 salads and 13 sandwiches.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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