Equation Alchemy Worksheet ALGEBRAIC BLUEPRINTS // SPEC-01
SYSTEM LEVEL: ALGEBRA 1 & 2
Project Title EQUATION ALCHEMY PRACTICE
DESIGNER (NAME) ___________________________
DATE // PERIOD ____ / ____ // ____
01
Structural Decomposition (Guided Walkthrough)
Before performing operations, decompose the equation's structure. Analyze the order of operations applied to the variable \(x\), then reverse them step-by-step.
TARGET ASSEMBLY: \(3\left(\frac{2x - 5}{4}\right) + 6 = 15\)
Step 1: Isolate term group Subtract 6 from both sides
Step 2: Clear outer scale Divide both sides by 3
Step 3: Eliminate divisor Multiply both sides by 4
Step 4: Extract variable Add 5, then divide by 2
02
Deconstruction Zone (Multi-Step Equations)
Deconstruct the following equations. Show your step-by-step mathematical reasoning. Keep equations balanced.
SPECIFICATION A \(4(3x - 2) = 2(x + 11)\)
FINAL VARIABLE VALUE: \(x =\) ________________
SPECIFICATION B \(\frac{5x - 3}{2} - 4 = 11\)
FINAL VARIABLE VALUE: \(x =\) ________________
03
Rational Resistors (Rational Equations)
Eliminate denominators by multiplying all terms by the Least Common Multiple (LCM). Check for extraneous solutions!
SPECIFICATION C \(\frac{3}{x - 2} + \frac{2}{x + 2} = \frac{12}{x^2 - 4}\)
EXTRANEOUS VALUES? \(x \neq\) ________________ FINAL SOLUTION: \(x =\) ________________
PROJECT NO: EP-L1-WS-01 PAGE 1 OF 2
Equation Alchemy // Part II
MODULE: LITERAL SYSTEMS
04
Literal Transmutations (Formula Rearrangements)
In physics and engineering design, you must isolate specific operational variables. Rearrange each formula below to solve for the target indicator.
SPECIFICATION D: KINETIC ENERGY \(E = \frac{1}{2}mv^2\)
Isolate the target velocity: \(v\)
RECONFIGURED FORMULA: \(v =\) ________________
SPECIFICATION E: THERMODYNAMICS \(P_1V_1T_2 = P_2V_2T_1\)
Isolate the initial temperature: \(T_1\)
RECONFIGURED FORMULA: \(T_1 =\) ________________
05
System Stress Test (Application Challenge)
An architectural beam's load capacity \(L\) (in kN) is given by the design formula: \(L = \frac{4w \cdot d^2}{L_{span}}\) where \(w\) is the width, \(d\) is the depth, and \(L_{span}\) is the span length.
CURRENT SPECS:
Width (\(w\)) = 5 cm
Span (\(L_{span}\)) = 12 m
Required Load (\(L\)) = 60 kN
CHALLENGE: Solve for the minimum required Depth (\(d\)).
DRAFTING BOARD // CALCULATIONS
REQUIRED DEPTH (d): \(d =\) ________________ cm
PROJECT NO: EP-L1-WS-01 PAGE 2 OF 2
Equation Alchemy Answer Key ALGEBRAIC BLUEPRINTS // TEACHER MASTER KEY
STATUS: APPROVED SOLUTION SET
Project Title EQUATION ALCHEMY SOLUTIONS
DESIGNER (NAME) Lenny (Master Key Architect)
DATE // PERIOD 17 AUG 2026 // ALL
01
Structural Decomposition (Guided Walkthrough)
Before performing operations, decompose the equation's structure. Analyze the order of operations applied to the variable \(x\), then reverse them step-by-step.
TARGET ASSEMBLY: \(3\left(\frac{2x - 5}{4}\right) + 6 = 15\)
Step 1: Isolate term group Subtract 6 from both sides
Step 2: Clear outer scale Divide both sides by 3
Step 3: Eliminate divisor Multiply both sides by 4
Step 4: Extract variable Add 5, then divide by 2
02
Deconstruction Zone (Multi-Step Equations)
Deconstruct the following equations. Show your step-by-step mathematical reasoning. Keep equations balanced.
SPECIFICATION A \(4(3x - 2) = 2(x + 11)\)
Step 1: Distribute outer scales
\(12x - 8 = 2x + 22\)
Step 2: Subract \(2x\) from both sides
\(10x - 8 = 22\)
Step 3: Add 8 to both sides
\(10x = 30\)
Step 4: Divide by 10
\(x = 3\)
VERIFIED MASTER VALUE: \(x = 3\)
SPECIFICATION B \(\frac{5x - 3}{2} - 4 = 11\)
Step 1: Isolate rational group (+4)
\(\frac{5x - 3}{2} = 15\)
Step 2: Eliminate divisor (*2)
\(5x - 3 = 30\)
Step 3: Isolate constant group (+3)
\(5x = 33\)
Step 4: Isolate target x (/5)
\(x = 33/5 = 6.6\)
VERIFIED MASTER VALUE: \(x = 6.6\) or \(\frac{33}{5}\)
03
Rational Resistors (Rational Equations)
Eliminate denominators by multiplying all terms by the Least Common Multiple (LCM). Check for extraneous solutions!
SPECIFICATION C \(\frac{3}{x - 2} + \frac{2}{x + 2} = \frac{12}{x^2 - 4}\)
LCM is \((x-2)(x+2) = x^2-4\)
Multiply all terms by \(x^2-4\):
\(3(x+2) + 2(x-2) = 12\)
\(3x + 6 + 2x - 4 = 12\)
Simplify expression:
\(5x + 2 = 12\)
\(5x = 10 \implies x = 2\)
ALERT: \(x=2\) makes denominator 0!
EXTRANEOUS VALUES: \(x = 2\) (makes division by 0) FINAL SOLUTION: No Solution (\(\emptyset\))
PROJECT NO: EP-L1-WS-01-KEY ANSWER KEY PAGE 1 OF 2
Equation Alchemy Key // Part II
MODULE: SOLUTIONS ARCHIVE
Boundary Builders Worksheet ALGEBRAIC BLUEPRINTS // SPEC-02
SYSTEM LEVEL: ALGEBRA 1 & 2
Project Title BOUNDARY BUILDERS WORKsheet
DESIGNER (NAME) ___________________________
DATE // PERIOD ____ / ____ // ____
01
Boundary Formulation (Technical Blueprint Guide)
Linear inequalities partition coordinate space into active zones. Use this technical specification checklist to construct boundaries accurately.
1. BOUNDARY LINE
Solid line for \(\le\) or \(\ge\) (boundary included). Dashed line for \(<\) or \(>\) (boundary excluded).
2. SHADING SECTOR
Pick test point \((0,0)\). If true, shade the region containing \((0,0)\). If false, shade the opposite side.
3. REGION OF VIABILITY
The overlap area in a system represents the Feasible Region containing all valid coordinate pairs.
02
Boundary Specs (Individual Plots)
Graph each individual boundary on the coordinate grids provided. Mark intercepts and shade the appropriate sector.
SPECIFICATION A \(y \ge -\frac{2}{3}x + 2\)
x y (-5, 5) (5, -5)
Line Type: ______________
Test Point (0,0): ______________
SPECIFICATION B \(3x - 4y > 12\)
x y (-5, 5) (5, -5)
Line Type: ______________
Test Point (0,0): ______________
PROJECT NO: EP-L2-WS-02 PAGE 1 OF 2
Boundary Builders // Part II
MODULE: SYSTEM CONSTRAINTS
03
System Constraints (Engineering Model)
An engineering company designs two drone variants: Recon (\(x\)) and Cargo (\(y\)). Production is limited by two critical assembly resources:
CONSTRAINT SPECIFICATION SHEET
Carbon Fiber Supply: Each Recon model uses 2 kg, while each Cargo model uses 4 kg. The total supply available is capped at 24 kg.
Assembly Labor: Each Recon model takes 3 hours, and each Cargo model takes 2 hours. The workshop labor limit is 18 hours.
Task 1: System Writing
Carbon Fiber Constraint Inequality: ________________________ \(\le\) 24
Labor Hours Constraint Inequality: ________________________ \(\le\) 18
Non-negativity Constraints: \(x \ge 0\), \(y \ge 0\)
Task 2: Feasibility Coordinates
Verify if the workshop can simultaneously build \(4\) Recon and \(3\) Cargo drones.
Coordinate Pair: \((4, 3)\) Carbon Fiber check: _______________ Labor Hours check: _______________
Is \((4,3)\) viable? [ ] YES // [ ] NO
CONSTRAINTS GRAPHING SYSTEM
Boundary Builders Answer Key ALGEBRAIC BLUEPRINTS // MASTER KEY SPEC-02
STATUS: VERIFIED PLOT SET
Project Title BOUNDARY BUILDERS SOLUTIONS
DESIGNER (NAME) Lenny (Master Key Architect)
DATE // PERIOD 17 AUG 2026 // ALL
01
Boundary Formulation (Technical Blueprint Guide)
Linear inequalities partition coordinate space into active zones. Use this technical specification checklist to construct boundaries accurately.
1. BOUNDARY LINE
Solid line for \(\le\) or \(\ge\) (boundary included). Dashed line for \(<\) or \(>\) (boundary excluded).
2. SHADING SECTOR
Pick test point \((0,0)\). If true, shade the region containing \((0,0)\). If false, shade the opposite side.
3. REGION OF VIABILITY
The overlap area in a system represents the Feasible Region containing all valid coordinate pairs.
02
Boundary Specs (Individual Plots)
Graph each individual boundary on the coordinate grids provided. Mark intercepts and shade the appropriate sector.
SPECIFICATION A \(y \ge -\frac{2}{3}x + 2\)
x y (-5, 5) (5, -5)
Line Type: SOLID (\(\ge\))
Test Point (0,0): FALSE (\(0 \ge 2\))
SPECIFICATION B \(3x - 4y > 12 \implies y < \frac{3}{4}x - 3\)
x y (-5, 5) (5, -5)
Line Type: DASHED (\(>\))
Test Point (0,0): FALSE (\(0 > 12\))
PROJECT NO: EP-L2-WS-02-KEY ANSWER KEY PAGE 1 OF 2
Boundary Builders Key // Part II
MODULE: SOLUTIONS MASTER
03
System Constraints (Engineering Model Solutions)
An engineering company designs two drone variants: Recon (\(x\)) and Cargo (\(y\)). Production is limited by two critical assembly resources:
CONSTRAINT SPECIFICATION SHEET
Carbon Fiber Supply: Each Recon model uses 2 kg, while each Cargo model uses 4 kg. The total supply available is capped at 24 kg.
Assembly Labor: Each Recon model takes 3 hours, and each Cargo model takes 2 hours. The workshop labor limit is 18 hours.
Task 1: System Writing
Carbon Fiber Constraint Inequality: \(2x + 4y \le 24\)
Labor Hours Constraint Inequality: \(3x + 2y \le 18\)
Non-negativity Constraints: \(x \ge 0\), \(y \ge 0\)
Task 2: Feasibility Coordinates
Verify if the workshop can simultaneously build \(4\) Recon and \(3\) Cargo drones.
Coordinate Pair: \((4, 3)\)
Carbon: \(2(4) + 4(3) = 8 + 12 = 20 \le 24\) [VALID]
Labor: \(3(4) + 2(3) = 12 + 6 = 18 \le 18\) [VALID]
Function Force Worksheet ALGEBRAIC BLUEPRINTS // SPEC-03
SYSTEM LEVEL: ALGEBRA 1 & 2
Project Title FUNCTION FORCE WORKSHEET
DESIGNER (NAME) ___________________________
DATE // PERIOD ____ / ____ // ____
01
Geometric Transformations (Force Mechanics)
The general vertex form describes how any parent function curve \(f(x)\) is shifted, scaled, or reflected in the coordinate plane: \(y = a \cdot f(x - h) + k\)
Vertical Scale (a) \(|a|>1\): Stretch
Reflection (-a) If \(a < 0\):
Horizontal (h) \(x - h\): Shift Right
Vertical (k) \(+k\): Shift Up
02
Deformation Mechanics (Quadratic & Absolute Value)
Deform the parent functions according to the specifications. Plot at least 5 integer coordinates for each curve.
SPECIFICATION A: QUADRATIC \(y = -(x - 2)^2 + 4\)
x y
Vertex Coordinate:( ____ , ____ )
Direction of Opening:_______________
Transformation Set:_______________
SPECIFICATION B: ABS VALUE \(y = 2|x + 1| - 3\)
x y
Vertex Coordinate:( ____ , ____ )
Direction of Opening:_______________
Transformation Set:_______________
PROJECT NO: EP-L3-WS-03 PAGE 1 OF 2
Function Force // Part II
MODULE: EXPONENTIAL DYNAMICS
03
Exponential Growth (Data Modeling)
In bioengineering, cell cultures double under optimal conditions. An exponential growth design curve is modelled by: \(y = 3 \cdot (2)^x\) where \(x\) represents hours and \(y\) represents million cells.
Task 1: Value Assembly
Evaluate the growth model at given intervals to assemble coordinate data points.
Hours (x) Cells (y) -1 ___________ 0 ___________ 1 ___________ 2 ___________ 3 ___________
Task 2: Structural Limits
Identify the boundary asymptote restricting exponential decay as \(x \to -\infty\).
Horizontal Asymptote Equation: \(y =\) ______________________
Y-Intercept Coordinate: ( _____ , _____ )
GROWTH PATHWAY PLOT
Function Force Answer Key ALGEBRAIC BLUEPRINTS // MASTER KEY SPEC-03
STATUS: APPROVED GRAPH PACK
Project Title FUNCTION FORCE SOLUTIONS
DESIGNER (NAME) Lenny (Master Key Architect)
DATE // PERIOD 17 AUG 2026 // ALL
01
Geometric Transformations (Force Mechanics)
The general vertex form describes how any parent function curve \(f(x)\) is shifted, scaled, or reflected in the coordinate plane: \(y = a \cdot f(x - h) + k\)
Vertical Scale (a) \(|a|>1\): Stretch
Reflection (-a) If \(a < 0\):
Horizontal (h) \(x - h\): Shift Right
Vertical (k) \(+k\): Shift Up
02
Deformation Mechanics (Quadratic & Absolute Value)
Deform the parent functions according to the specifications. Plot at least 5 integer coordinates for each curve.
SPECIFICATION A: QUADRATIC \(y = -(x - 2)^2 + 4\)
x y
Vertex Coordinate:( 2 , 4 )
Direction of Opening:DOWNWARD (Reflected)
Transformation Set:Right 2, Up 4, Reflect X
SPECIFICATION B: ABS VALUE \(y = 2|x + 1| - 3\)
x y
Vertex Coordinate:( -1 , -3 )
Direction of Opening:UPWARD
Transformation Set:Left 1, Down 3, V-Stretch *2
PROJECT NO: EP-L3-WS-03-KEY ANSWER KEY PAGE 1 OF 2
Function Force Key // Part II
MODULE: EXPONENTIAL ANSWERS
03
Exponential Growth (Data Modeling Solutions)
In bioengineering, cell cultures double under optimal conditions. An exponential growth design curve is modelled by: \(y = 3 \cdot (2)^x\) where \(x\) represents hours and \(y\) represents million cells.
Task 1: Value Assembly
Evaluate the growth model at given intervals to assemble coordinate data points.
Hours (x) Cells (y) -1 \(3 \cdot (2)^{-1} = 1.5\) 0 \(3 \cdot (1) = 3\) 1 \(3 \cdot (2) = 6\) 2 \(3 \cdot (4) = 12\) 3 \(3 \cdot (8) = 24\)
Task 2: Structural Limits
Identify the boundary asymptote restricting exponential decay as \(x \to -\infty\).
Horizontal Asymptote Equation: \(y = 0\) (The x-axis limits decay)
