Friday
Friday: Systems All Around
Rocket times himself crossing the garage. "I walk 4 tiles a second. Raven, you get a 6 tile head start at 2 a second."
"Distance equals rate times time," Raven says. "Mine is d = 2t + 6. Yours is d = 4t. What do you notice?"
"Two equations, two unknowns," Rocket says. "4t = 2t + 6. So t = 3, and d = 12. I catch you at tile 12."
Nova hovers over tile 12, her light marking the spot. "And if you both walked 4 tiles a second?"
"d = 4t and d = 4t + 6. Parallel. I never catch up," Rocket says.
"Same slope, different start," Raven says. "No solution, just like last week's card."
Rocket grins. "Puzzle Night needs a catch-up race. Everyone will want to bet on the crossing."
"Everyone will want to check it," Raven says. "Review first."
Where systems live
A catch-up race: d = 4t and d = 2t + 6 cross at (3, 12). Rocket catches up after 3 seconds, at tile 12.
Two walkers at the same speed: d = 4t and d = 4t + 6 are parallel. The gap stays 6 tiles. No solution.
Two timers that total 30 minutes and differ by 6: a total fact and a difference fact make a system.
Two jugs filling at different rates from different starts: the equal-level moment is a crossing point.
The week in five lines
- A system is two equations that must both be true. Its solution is the point where the lines cross.
- Inspection spots no solution (same expression, different numbers) and infinitely many (the same line twice).
- Graphing estimates the crossing. Substitution finds it exactly.
- Same slope and different intercepts means parallel lines and no solution.
- A puzzle with two unknowns needs two facts, one equation each.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| Parallel lines have the same slope and never cross. | ? |
| y = 4x and y = 4x + 6 have no solution. | ? |
| y = 4x and y = 2x + 6 have one solution. | ? |
| (3, 12) is a solution of y = 4x and y = 2x + 6. | ? |
| One equation is enough to find two unknown numbers. | ? |
Measure the water height every minute. Write both rules, predict the level moment, then watch for it.
Keep water over a tray or the sink, and wipe spills right away.
Terrific week. You can solve two equations at once three different ways. Tomorrow is Puzzle Day.