➕ Algebra 1 · Systems of Equations
Systems Word Problems: true or false
Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.
All statements and answers
If a story gives only one relationship between two unknown quantities, such as x + y = 20, that is not enough to find a single pair of answers.
✓ True. x + y = 20 is satisfied by (10, 10), (15, 5), (0, 20) and many other pairs. You need a second fact.
A problem says 90 tickets were sold in all, at 4 dollars and 7 dollars each. The equation 4s + 7a = 90 says that 90 tickets were sold.
✗ False. The 90 is a number of tickets, so the count equation is s + a = 90. The expression 4s + 7a is a number of dollars.
A pair that satisfies both of your equations can still be wrong for the story if you mistranslated a sentence.
✓ True. The check in your equations tests only your algebra. If an equation does not match the story, the pair can satisfy the equations and still be wrong.
If the algebra gives 6.5 student tickets, you should round to 7 tickets and report that.
✗ False. Rounding changes the totals, so the rounded numbers no longer fit the story. Recheck the setup. If it is right, the situation as described is impossible.
“Let x be tickets” is not a clear definition, because it does not say whether x is a number of tickets, a price or a total cost.
✓ True. A good definition says what the letter measures, with its unit: “let x be the number of adult tickets sold.”
The plans C = 25 + 7t and C = 16 + 7t never cost the same.
✓ True. Setting them equal gives 25 = 16, which is false. The rates match, so the 9-dollar gap in the fees never changes.
Two plans with different monthly rates must cost the same after some number of months t ≥ 0.
✗ False. Take C = 10 + 5t and C = 20 + 8t. Setting them equal gives t = −10/3, which is not a possible number of months. The second plan starts higher and grows faster, so it costs more for every t ≥ 0.
On a graph of cost in dollars (vertical) against months (horizontal), two plan lines cross at (5, 80). The plans cost the same, 80 dollars, after 5 months.
✓ True. A point is written (x, y), so here the first number is months and the second is dollars. At the crossing both plans have the same cost.
A box holds d dimes and q quarters worth 260 cents in all, so d + q = 260.
✗ False. d + q counts coins. The 260 cents is a value, so the value equation is 10d + 25q = 260.
If correct equations for a story give −3 cars, the situation as described is impossible.
✓ True. A number of cars cannot be negative. The equations do not know that, so the story check has to catch it. (If the equations might be mistranslated, recheck them first.)
Solving a system word problem by substitution and by elimination can give two different correct answers.
✗ False. Both methods find the solution of the same system, so a correct solution is the same either way.
A good final answer says what each number counts, with units, in a sentence, such as “There were 8 small boxes and 3 large boxes.”
✓ True. The sentence tells the reader which number goes with which quantity and what the unit is. A bare pair of numbers does not.