Scythe Systems Slides
SCYTHE SYSTEMS
Solving Linear Equations in the Underworld
"Billy and Mandy have scrambled the Soul Records! Now the numbers don't add up."
The Underworld Crisis
Grim has two variables to track: Souls (x) and Scythe Repairs (y).
To fix the records, we must solve Systems of Equations using two specific dark arts:
- The Substitution Ritual
- The Elimination Curse
Substitution Method
Step 1: Isolate
Solve one equation for either \(x\) or \(y\).
Step 2: Plug In
Substitute that expression into the other equation.
Step 3: Solve & Back-Solve
Solve for the variable, then find the second one.
Best used when a variable is already alone (coefficient of 1)!
Substitution Example
\[ y = 2x - 3 \] \[ 3x + y = 12 \]
Notice \(y\) is already isolated in the first equation!
1. Plug it in:
\( 3x + (2x - 3) = 12 \)
2. Solve for x:
\( 5x - 3 = 12 \rightarrow 5x = 15 \rightarrow x = 3 \)
3. Find y:
\( y = 2(3) - 3 \rightarrow y = 3 \)
Elimination Method
Step 1: Align
Stack variables and constants (Standard Form).
Step 2: Match & Negate
Multiply equations so one variable has opposite coefficients.
Step 3: Add & Solve
Add the equations to "eliminate" one variable.
\( +1x \) and \( -1x \)
Perfect for when equations are already lined up!
Elimination Example
\[ 2x + y = 10 \] \[ 2x - y = 6 \]
Notice the \(+y\) and \(-y\) are ready to disappear!
1. Add them up:
\( (2x + 2x) + (y - y) = 10 + 6 \)
2. Solve for x:
\( 4x = 16 \rightarrow x = 4 \)
3. Find y:
\( 2(4) + y = 10 \rightarrow 8 + y = 10 \rightarrow y = 2 \)
Decoding the Underworld
Define Variables
What are the two unknown values?
Write Equations
Look for totals or comparisons in the text.
Pick a Method
Which path is easier? Isolate or Eliminate?
"Don't let Billy touch the calculator..." - Grim
Soul Record Worksheet
Soul Record Recovery
Help Grim fix Billy and Mandy's mathematical mess!
Student:
Date:
Part I: The Substitution Ritual
Isolate one variable and plug it into the other equation to solve.
\[ y = 4x - 1 \] \[ 2x + y = 5 \]
Show Your Work
Answer:
\[ x + 2y = 10 \] \[ 3x - y = 9 \]
Show Your Work
Answer:
Part II: The Elimination Curse
Add or subtract the equations to eliminate one variable. You may need to multiply first!
\[ 5x + y = 13 \] \[ 2x - y = 1 \]
Show Your Work
Answer:
\[ 2x + 3y = 12 \] \[ 4x - y = 10 \]
Show Your Work
Answer:
Part III: Underworld Chaos
5
Mandy is counting the inventory. She finds that the number of Potions (p) and Curses (c) totals 25. If there are 7 more Curses than Potions, how many of each does Grim have in stock?
Set Up Equations
Solve Here
6
Billy went to the Underworld snack bar. He bought 3 Brain Tacos (t) and 2 Ecto-Sodas (s) for $19. Mandy bought 2 Brain Tacos and 4 Ecto-Sodas for $22. What is the price of a single Brain Taco and a single Ecto-Soda?
Set Up Equations
Solve Here
Soul Record Answer Key
ANSWER KEY
Soul Record Recovery: Systems of Equations
Teacher Resource
Part I: Substitution
1. \( y = 4x - 1 \); \( 2x + y = 5 \)
2x + (4x - 1) = 5
6x - 1 = 5
6x = 6 → x = 1
y = 4(1) - 1 = 3
Answer: (1, 3)
2. \( x + 2y = 10 \); \( 3x - y = 9 \)
x = 10 - 2y
3(10 - 2y) - y = 9
30 - 6y - y = 9
-7y = -21 → y = 3
x = 10 - 2(3) = 4
Answer: (4, 3)
Part II: Elimination
3. \( 5x + y = 13 \); \( 2x - y = 1 \)
(5x + 2x) + (y - y) = 13 + 1
7x = 14 → x = 2
5(2) + y = 13
y = 3
Answer: (2, 3)
4. \( 2x + 3y = 12 \); \( 4x - y = 10 \)
Multiply 2nd eq by 3: 12x - 3y = 30
(2x + 12x) + (3y - 3y) = 12 + 30
14x = 42 → x = 3
4(3) - y = 10 → y = 2
Answer: (3, 2)
Part III: Word Problems
5. Potions (p) and Curses (c)
p + c = 25
c = p + 7
p + (p + 7) = 25
2p = 18 → p = 9
c = 9 + 7 = 16
9 Potions, 16 Curses
6. Tacos (t) and Sodas (s)
3t + 2s = 19
2t + 4s = 22
Mult 1st by -2: -6t - 4s = -38
-4t = -16 → t = 5
3(5) + 2s = 19 → 2s = 4 → s = 2
Tacos: $5, Sodas: $2
