Week 06 · Systems of Equations and Inequalities
Loft Day
Loft Day: the family catch-up race
// Two carts meet, and a region of good builds
⏱ about 20 min
Loft Day: The Family Catch-Up Race
Saturday the family lines up in the backyard for a walking race with a head start.
"Rules," Comet says. "The slow walker starts 6 meters ahead and walks 1 meter a second. The fast walker does 2."
"Predict the catch-up spot before anyone moves," Wren says. "What do you notice?"
"y = x + 6 and y = 2x," says Comet's cousin. "2x = x + 6, so x is 6. Six seconds, 12 meters."
Nova projects a line on the grass at 12 meters. "Would you like a hint? Count out loud as you walk."
The fast walker draws level right at the projected line. Everyone cheers.
"Now you pick the head start and the speeds," Wren says, handing the chalk to the youngest.
- A system is two equations that must both be true. Its solution is the crossing point of the two lines.
- Solve by graphing, substitution or elimination. Elimination works because adding true equations keeps them true.
- Same slope and different intercepts: parallel, no solution. The same line twice: infinitely many.
- The x of a crossing is the solution of f(x) = g(x).
- An inequality shades a half-plane. Constraints on a build make a region, and viable options sit inside it.
◇ FAMILY CATCH-UP RACE
Mark a start line and a head-start line with chalk or tape. Pick a slow speed and a fast speed, in steps per second.
Write both rules and solve the system together to predict the catch-up point. Mark it before anyone walks.
Walk the race, counting seconds out loud. Did the fast walker draw level at the mark?
Trade roles and change the head start. Try two equal speeds and watch the gap stay the same.
Write both rules and solve the system together to predict the catch-up point. Mark it before anyone walks.
Walk the race, counting seconds out loud. Did the fast walker draw level at the mark?
Trade roles and change the head start. Try two equal speeds and watch the gap stay the same.
For a grown-up
Walk, never run, on a clear flat stretch with no obstacles. Keep the race away from streets and driveways.
If the prediction misses, that is a good moment: compare the counted seconds with the rule and ask what changed.
Chalk washes off. Tape comes up afterwards so no one trips.
If the prediction misses, that is a good moment: compare the counted seconds with the rule and ask what changed.
Chalk washes off. Tape comes up afterwards so no one trips.
LOFT DAY CHECK
- Read the question.
- Tap your answer.
The slow walker is y = x + 6 and the fast walker is y = 2x. Where do they meet?A head start of 8 at 1 a second against 3 a second: y = x + 8 and y = 3x. Where do they meet?
Both walkers at 1 a second, one 6 ahead: y = x + 6 and y = x. How many solutions?
The family race was y = x + 6 and y = 2x. After how many seconds did the fast walker catch up? Type the number.
Set the rules equal: 2x = x + 6. Solve for x.
WHY THIS EXERCISEA prediction becomes a solution only when it passes both equations and the real race.
Draw the family race as two lines, seconds across and meters up. Mark the catch-up point.