Vault Breaker Worksheet Vault Breaker
MISSION: Solve the Systems using Substitution
Agent Name:
Date/Log:
Security Briefing: Substitute one equation into the other to eliminate a variable. Solve for the remaining variable, then plug it back in to find the full coordinate pair \((x, y)\).
CODE 01
\[ \begin{cases} y = 2x - 3 \\ x + y = 9 \end{cases} \]
Solution:
CODE 02
\[ \begin{cases} x = y + 5 \\ 2x - y = 7 \end{cases} \]
Solution:
CODE 03
\[ \begin{cases} y = -3x + 1 \\ 2x - y = 9 \end{cases} \]
Solution:
CODE 04
\[ \begin{cases} x = 4y \\ x - 2y = -6 \end{cases} \]
Solution:
CODE 05
\[ \begin{cases} y = x - 4 \\ 3x + 2y = 7 \end{cases} \]
Solution:
CODE 06
\[ \begin{cases} x = -2y + 1 \\ 3x + 4y = -1 \end{cases} \]
Solution:
CODE 07
\[ \begin{cases} y = 5x - 7 \\ x + 2y = -3 \end{cases} \]
Solution:
CODE 08
\[ \begin{cases} x = y - 8 \\ 2x + 3y = 14 \end{cases} \]
Solution:
CODE 09
\[ \begin{cases} y = -x \\ 4x - 3y = 14 \end{cases} \]
Solution:
CODE 10
\[ \begin{cases} x = 2y + 10 \\ x + 3y = 5 \end{cases} \]
Solution:
Cyber Maze Bypass Activity Cyber Maze Bypass
MISSION: Find the correct path to the core by solving the systems.
Agent:
Bypass Protocol:
Start at the "ENTRY" node. Solve the system of equations. Follow the path matching your solution \((x, y)\) to the next node. Continue until you reach the "CORE". Only one path leads to the final system override!
START
\[ \begin{cases} y = 2x \\ x + y = 6 \end{cases} \]
(2, 4)
(3, 3)
\[ \begin{cases} x = y - 3 \\ 2x + y = 6 \end{cases} \]
Dead End
(1, 4)
(4, 7)
\[ \begin{cases} y = -x + 5 \\ 3x - y = 7 \end{cases} \]
(3, 2)
(2, 3)
\[ \begin{cases} x = 2y - 1 \\ x + y = 8 \end{cases} \]
Signal Lost
(5, 3)
SYSTEM CORE
BYPASS SUCCESS
Authorized Personnel Only
Signal Processing Area (Work Space) Substitution Systems Answer Key Vault Breaker: Answer Key
Teacher Resource | Substitution Method
CODE 01
(4, 5)
Work: x + (2x - 3) = 9 → 3x = 12 → x = 4; y = 2(4) - 3 = 5
CODE 02
(2, -3)
Work: 2(y + 5) - y = 7 → y + 10 = 7 → y = -3; x = -3 + 5 = 2
CODE 03
(2, -5)
Work: 2x - (-3x + 1) = 9 → 5x - 1 = 9 → x = 2; y = -3(2) + 1 = -5
CODE 04
(-12, -3)
Work: 4y - 2y = -6 → 2y = -6 → y = -3; x = 4(-3) = -12
CODE 05
(3, -1)
Work: 3x + 2(x - 4) = 7 → 5x - 8 = 7 → x = 3; y = 3 - 4 = -1
CODE 06
(-3, 2)
Work: 3(-2y + 1) + 4y = -1 → -2y + 3 = -1 → -2y = -4 → y = 2; x = -2(2) + 1 = -3
CODE 07
(1, -2)
Work: x + 2(5x - 7) = -3 → 11x - 14 = -3 → 11x = 11 → x = 1; y = 5(1) - 7 = -2
CODE 08
(-2, 6)
Work: 2(y - 8) + 3y = 14 → 5y - 16 = 14 → 5y = 30 → y = 6; x = 6 - 8 = -2
CODE 09
(2, -2)
Work: 4x - 3(-x) = 14 → 7x = 14 → x = 2; y = -2
CODE 10
(8, -1)
Work: (2y + 10) + 3y = 5 → 5y = -5 → y = -1; x = 2(-1) + 10 = 8
Cyber Maze: Answer Path
START (ENTRY)
(2, 4)
(1, 4)
(3, 2)
(5, 3)
CORE
