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Algebra 1 9-12 / Week 06 / Friday
5/6
Week 06 · Systems of Equations and Inequalities

Friday

Systems all around
// Two carts meet, and a region of good builds
⏱ about 20 min

Friday: Systems All Around

Comet drops onto the Loft couch. "Systems are everywhere now. Two timers, two jugs, two carts."

"Two jugs," Wren says. "One at 200 milliliters rising 40 a minute. One at 100 rising 60. When are they level?"

"Set them equal: 40x + 200 = 60x + 100. So 100 = 20x and x is 5." Comet nods. "Five minutes, 400 each."

Nova projects the two jug lines crossing at (5, 400). "Would you like a hint? That x is where the graphs meet."

"The x of the crossing solves f(x) = g(x)," Wren says. "What do you notice about the two timers card?"

"Total 30 minutes, difference 6. x + y = 30 and x - y = 6. Elimination!" Comet says.

"Review first," Wren says. "Our engineer, then next week the strip folds and the ball bounces."

Where systems live

  • Two jugs: y = 40x + 200 and y = 60x + 100 are level at (5, 400). Five minutes, 400 milliliters each.
  • Two timers that total 30 minutes and differ by 6: a total fact and a difference fact make a system.
  • A catch-up race: a head start against a faster speed. The crossing is the moment of catching up.
  • Build limits: at most so many parts, at least so many of one kind. Inequalities shade the good region.
A grid from 0 to 10 with the two jug lines crossing at a marked point, drawn in hundreds of milliliters.

The week in five lines

  • A system is two equations that must both be true. Its solution is the point where the lines cross.
  • The graph of an equation is all its solutions. The x of a crossing solves f(x) = g(x).
  • Solve by graphing (estimate), substitution (y = something) or elimination (add a multiple of one equation).
  • Elimination keeps the solutions because adding true equations gives a true equation.
  • An inequality shades a half-plane. A system of inequalities shades the overlap. Constraints pick viable options.
EVERYDAY SYSTEMS
  • Read the question.
  • Tap your answer.
Jug A is y = 40x + 200 and jug B is y = 60x + 100. Which (x, y) is the level moment?
Setting the jugs equal gives 40x + 200 = 60x + 100. After how many minutes are they level?
Two timers total 30 minutes and differ by 6. Which pair (x, y) fits x + y = 30 and x - y = 6?
Two jugs both rising 40 a minute from 200 and 100: y = 40x + 200 and y = 40x + 100. How many solutions?
MIXED REVIEW
  • Read the question.
  • Tap your answer.
Which point makes both y = x + 4 and y = 2x true?
x + y = 24 and x - y = 4. Add the two equations. What equation is left?
A small build: x + y ≤ 12 and y ≥ x. Which option is viable?
Is (3, 2) a solution of y ≥ x?
StatementTrue or false?
Parallel lines have the same slope and never cross.?
y = 40x + 200 and y = 40x + 100 have no solution.?
y = 40x + 200 and y = 60x + 100 have one solution.?
(5, 400) is a solution of y = 40x + 200 and y = 60x + 100.?
A dashed boundary line means points on the line are solutions.?
(7, 6) satisfies x + y ≤ 12 and y ≥ x.?
WHY THIS EXERCISEThese lines are the whole week. Next week the strip doubles and the ball bounces: exponential change.
Try it
Set two slow drips over a tray, one starting with more water. Measure every minute, write both rules, predict the level moment.
Keep water over the tray or sink and wipe spills right away.
Draw the two jugs as lines, minutes across and milliliters up. Circle the level moment.

Terrific week. You can solve systems three ways and shade a region of good builds. Tomorrow is Loft Day.

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