Multiply Eq 1 by a tactical scalar to create opposite \(y\)-terms:
\(2x - y = 7\)
\(5x + 3y = 23\)
Multiplier: Multiply Eq 1 by \(3\) TARGET: \((x, y)\)
CARD E-03 // ELIMINATION 5 PTS
Double Scalar: Scale both equations to eliminate either term:
\(3x + 2y = 1\)
\(4x + 5y = 6\)
Find LCM for coefficients TARGET: \((x, y)\)
CARD E-04 // ELIMINATION 5 PTS
Zero-Step Neutralizer: Directly eliminate \(y\) and verify:
\(4x - 3y = 17\)
\(2x + 3y = 13\)
Sum equations → solve for \(x\) TARGET: \((x, y)\)
CARD M-01 // REAL-LIFE MODEL 10 PTS
The Premiere Tickets: The theater sold 150 total tickets. Student tickets cost $5 each and adult tickets cost $9 each. Total revenue was $1,070.
Define variables • Set up system • Solve & state units
Strategy: Elimination or Substitution? TICKETS: \((s, a)\)
CARD M-02 // REAL-LIFE MODEL 10 PTS
Cyber Subscriptions: Plan Alpha costs a $15 startup fee plus $8/month. Plan Beta costs a $27 startup fee plus $5/month. At how many months are costs equal?
Write two cost equations \(C = mx + b\) • Find break-even
Strategy: Why is setting equal optimal? SOLVE: \((m, C)\)
CARD M-03 // REAL-LIFE MODEL 10 PTS
Drone Cargo Payload: A drone carries 30 total sensor pods. Thermal pods weigh 3 lbs each; signal pods weigh 5 lbs each. The total cargo weighs 114 lbs.
Eq 1: Total pod count • Eq 2: Total payload weight
Which variable is easiest to eliminate? PODS: \((t, s)\)
CARD M-04 // REAL-LIFE MODEL 10 PTS
Command Perimeter: A rectangular compound has a perimeter of 56 meters (\(2l + 2w = 56\)). Its length \(l\) is 4 meters less than three times its width \(w\).
Translate: \(l = 3w - 4\) • Determine width & length
Strategy: Why is Substitution fastest? DIMS: \((w, l)\)
SECTOR 3 & 4 CHALLENGE CARDS • KEEP CARDS WITH CODED MISSION SQUAD
Eq 2: __________________________________________________
Chosen Strategy: ( ) Graphing ( ) Substitution ( ) Elimination
Tactical Rationale: __________________________________________________
Final Real-World Interpretation:
Ans: ___________________________________________________________
Mission Card: M-_____ 10 Base + Strategy Bonus
Let variable ______ \(=\) ________________________________________
Let variable ______ \(=\) ________________________________________
Formulated System of Equations:
Eq 1: __________________________________________________
Eq 2: __________________________________________________
Chosen Strategy: ( ) Graphing ( ) Substitution ( ) Elimination
Tactical Rationale: __________________________________________________
Final Real-World Interpretation:
Ans: ___________________________________________________________
Squad Mission Score Summary Sec 1 (10) + Sec 2 (10) + Sec 3 (10) + Sec 4 (20) + Bonuses (10)
Base Pts _____ / 50
Strategy Bonus _____ / 10
Mission Total _____ / 60
PAGE 2 OF 2 // OPERATION INTERCEPT SQUAD LOG // SUBMIT TO COMMANDER UPON ROTATION COMPLETION
Top Secret // Answer Key
TEACHER ONLY
Master solution breakdown for Sector 3 (Elimination) & Sector 4 (Real-World Modeling).
3
CARD E-01 (2, 3)
Add equations directly: \((3x - 3x) + (4y + 2y) = 18 + 0\)
\(6y = 18 \implies y = 3\). Substitute back: \(3x + 4(3) = 18 \implies x = 2\).
CARD E-02 (4, 1)
Multiply Eq 1 by \(3\): \(6x - 3y = 21\). Add to Eq 2:
\(11x = 44 \implies x = 4\). Substitute: \(2(4) - y = 7 \implies y = 1\).
CARD E-03 (-1, 2)
Scale Eq 1 by \(5\) (\(15x+10y=5\)) and Eq 2 by \(-2\) (\(-8x-10y=-12\)):
Sum: \(7x = -7 \implies x = -1\). Substitute: \(3(-1) + 2y = 1 \implies y = 2\).
CARD E-04 (5, 1)
Add directly: \((4x+2x) + (-3y+3y) = 17 + 13 \implies 6x = 30\)
\(x = 5\). Substitute back: \(2(5) + 3y = 13 \implies 3y = 3 \implies y = 1\).
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CARD M-01 // TICKETS 70 student, 80 adult
Variables: \(s\) = student tickets, \(a\) = adult tickets
System: \(s + a = 150\) and \(5s + 9a = 1070\)
Strategy: Elimination (scale Eq 1 by \(-5\)) or Substitution (\(s = 150-a\)).
Work: \(5(150-a) + 9a = 1070 \implies 750 + 4a = 1070 \implies 4a = 320 \implies a = 80, s = 70\).
CARD M-02 // SUBSCRIPTIONS 4 months, $47 total
Variables: \(m\) = months, \(C\) = total cost in dollars
System: \(C = 8m + 15\) and \(C = 5m + 27\)
Strategy: Substitution (Equal values): Set \(8m + 15 = 5m + 27\).
Work: \(3m = 12 \implies m = 4\). Cost \(C = 8(4) + 15 = \$47\).
CARD M-03 // DRONE CARGO 18 thermal, 12 signal
Variables: \(t\) = thermal pods (3 lbs), \(s\) = signal pods (5 lbs)
System: \(t + s = 30\) and \(3t + 5s = 114\)
Strategy: Elimination (scale Eq 1 by \(-3\)): \(-3t - 3s = -90\).
Work: Adding gives \(2s = 24 \implies s = 12\) signal pods, \(t = 18\) thermal pods.
CARD M-04 // PERIMETER Width = 8m, Length = 20m
Variables: \(w\) = width (meters), \(l\) = length (meters)
System: \(2l + 2w = 56\) (or \(l + w = 28\)) and \(l = 3w - 4\)
Strategy: Substitution: \(l\) is already isolated.
Work: \(2(3w - 4) + 2w = 56 \implies 8w - 8 = 56 \implies 8w = 64 \implies w = 8\text{ m}, l = 20\text{ m}\).
Critical Facilitator Intervention Points (Watch For These Common Errors):
1. Distribution Negatives: Students forgetting to distribute negative signs (e.g., in Card S-04, \(-2(5 - x)\) becoming \(-10 - 2x\) instead of \(-10 + 2x\)).
2. Missing Variables: In word problems, students writing numbers without declaring what \(x\) and \(y\) represent. Enforce "Let" statements!
3. Half-Solved Coordinates: Students stopping after solving for one variable and forgetting to substitute back to find the complete \((x, y)\) coordinate.
PAGE 2 OF 2 // MASTER SOLUTION DOSSIER // OPERATION INTERCEPT VERIFICATION COMPLETE
