The Pruner's Practice

Edition 11, Lesson 11.2: Elimination

Part I: The Natural Opposite

In these problems, the threads are already prepared for pruning. Simply add them to eliminate one variable.

1. The First Snip:
Thread A: 3x + y = 14
Thread B: 2x - y = 6
Solution (x, y): ________
2. The Balanced Branch:
Thread A: 5x + 3y = 20
Thread B: -5x + y = -4
Solution (x, y): ________
3. The Hidden Fruit:
Thread A: 4x - 2y = 8
Thread B: 3x + 2y = 13
Solution (x, y): ________
The Gardener's Check:

When you added the equations, did one variable completely disappear (become 0)? If not, check your signs!

Part II: Preparing the Branch (Scaling)

In these problems, you must multiply one equation by a number (like -1 or 2) to create opposites.

4. Identical Threads:
Thread A: 2x + 5y = 15
Thread B: 2x + y = 7
(Hint: Multiply Thread B by -1)
Solution (x, y): ________
5. The Growing Branch:
Thread A: x + 2y = 10
Thread B: 3x - 4y = 10
(Hint: Multiply Thread A by 2)
Solution (x, y): ________
6. Targeting the 'x':
Thread A: 2x + 3y = 12
Thread B: 4x - y = 10
(Hint: Multiply Thread A by -2)
Solution (x, y): ________

Part III: The Double Pruning (Advanced)

In these problems, you must multiply BOTH equations to find a common factor for elimination.

7. The Weaver's Knot:
Thread A: 3x + 2y = 11
Thread B: 2x + 3y = 14
(Hint: Multiply Thread A by 2 and Thread B by -3 to eliminate 'x')
Solution (x, y): ________
8. The Vineyard's Challenge:
Thread A: 5x - 4y = 2
Thread B: 3x + 3y = 21
(Hint: Multiply Thread A by 3 and Thread B by 4 to eliminate 'y')
Solution (x, y): ________

Part IV: Literal Pruning (The Algebra of Form)

Elimination works even when the truth is hidden behind letters. Use the Gardener's Logic to solve for 'x' in this abstract system.

9. The Pure Branch:
Thread 1: ax + y = c
Thread 2: bx - y = d
x = ____________________

Part V: Kingdom Modeling

10. The Shared Harvest:
Two workers gather wheat (w) and barley (b). Worker 1 gathers 5 bushels of wheat and 2 bushels of barley for 31 coins. Worker 2 gathers 3 bushels of wheat and 2 bushels of barley for 21 coins.

Rule 1: 5w + 2b = 31
Rule 2: 3w + 2b = 21

Use elimination to find the value of each grain.
Wheat: ____ Barley: ____
11. The Shepherd's Count:
A shepherd has sheep and goats. If he adds his sheep (s) and his goats (g), he has 100 animals. If he subtracts the number of goats from twice the number of sheep, he has 80.

Rule 1: s + g = 100
Rule 2: 2s - g = 80

How many of each does he have?
Sheep: ____ Goats: ____

Part VI: Logic and Reflection

12. The Law of the Entire Branch:
Look at this attempt at elimination:
2(x + 3y = 10) -> 2x + 6y = 10

What error did the student make? Why is this a "Rupture" in the Gardener's Logic?
13. Choosing the Tool:
Compare Lesson 11.1 (Substitution) and 11.2 (Elimination). When is it better to use the Weaver's method (Substitution)? When is it better to use the Gardener's method (Elimination)?
14. Final Reflection on the Gardener:
Read John 15:1-2. How does the mathematical act of "Elimination" change your perspective on what God is doing when He prunes things from your life? Is it an act of destruction or an act of revelation?
"I vow to respect the entire branch, applying my operations to both sides of the truth. I will not seek shortcuts that bypass the Gardener's precision, but will find joy in the clarity of the result. I welcome the pruning that leads to a more fruitful understanding of the Father's world."

Part VII: The Gardener's Advanced Practice

These problems require multiple steps and careful attention to signs.

15. The Triple Challenge:
Thread A: 3x + 5y = 29
Thread B: 2x + 3y = 18
(Hint: Scale both equations to eliminate x or y)
Solution (x, y): ________
16. The Negative Garden:
Thread A: -2x + 4y = 10
Thread B: 3x - 4y = -5
Solution (x, y): ________
17. The Large Coefficients:
Thread A: 5x + 6y = 32
Thread B: 3x + 4y = 20
Solution (x, y): ________
18. The Fractional Result:
Thread A: 2x + 3y = 11
Thread B: 4x + 5y = 19
Solution (x, y): ________

Part VIII: Special Cases in Elimination

What happens when the system has no solution or infinite solutions?

19. The Dependent Vine:
Thread A: 2x + y = 8
Thread B: 4x + 2y = 16
What happens when you eliminate? ________________________________
System Type: _____________ | Number of Solutions: _____________
20. The Parallel Paths:
Thread A: 3x + 2y = 12
Thread B: 6x + 4y = 30
What happens when you eliminate? ________________________________
System Type: _____________ | Number of Solutions: _____________

Part IX: Kingdom Modeling Extended

21. The Merchant's Scales:
At the market, 4 loaves of bread and 3 jars of honey cost 31 coins.
Also, 2 loaves of bread and 3 jars of honey cost 19 coins.

Rule 1: 4b + 3h = 31
Rule 2: 2b + 3h = 19

Find the cost of one loaf and one jar of honey.
Bread: ____ coins | Honey: ____ coins
22. The Builder's Inventory:
3 cedar boards and 5 oak boards cost 54 coins.
4 cedar boards and 2 oak boards cost 36 coins.

Rule 1: 3c + 5o = 54
Rule 2: 4c + 2o = 36

Find the cost of each type of board.
Cedar: ____ coins | Oak: ____ coins
23. The Shepherd's Profit:
Selling 5 sheep and 3 goats brings 87 coins.
Selling 2 sheep and 4 goats brings 52 coins.

What is the selling price of one sheep and one goat?
Sheep: ____ coins | Goat: ____ coins
24. The Potter's Clay:
Red clay costs 'r' coins per pound. Blue clay costs 'b' coins per pound.
6 pounds of red and 4 pounds of blue cost 44 coins.
3 pounds of red and 8 pounds of blue cost 50 coins.

Find the cost per pound of each clay.
Red: ____ coins/lb | Blue: ____ coins/lb

Part X: Method Comparison

25. Substitution vs. Elimination:
Solve this system using BOTH methods. Then answer: which was easier for this problem?

Thread A: x + y = 10
Thread B: x - y = 4

Using Substitution:

Using Elimination:

Solution: (x, y) = ________
Which method was easier for this problem? _______________________
Why? ________________________________________________

Part XI: The Glossary Check

Define each term in your own words.

26. Elimination Method:
27. Opposite Coefficients:
28. Scaling:
29. Standard Form (Ax + By = C):

Part XII: Reflection Journal

30. The Gardener's Heart:
Read John 15:1-2. "I am the true vine, and my Father is the gardener. He cuts off every branch in me that bears no fruit, while every branch that does bear fruit he prunes so that it will be even more fruitful."

How does the mathematical act of "elimination" reflect God's work in our lives? When God "prunes" something from your life, is it destruction or revelation?
31. My Pruning Story:
Think of a time when something was "eliminated" from your life—a habit, a relationship, an opportunity. At first, did it feel like loss? Looking back, did it bring clarity or new growth?
The Gardener's Final Check:

Before closing this workbook, verify that:

"I vow to respect the entire branch, applying my operations to both sides of the truth. I will not seek shortcuts that bypass the Gardener's precision, but will find joy in the clarity of the result. I welcome the pruning that leads to a more fruitful understanding of the Father's world. I am ready for the Watchman's lesson."
Answer Key Reference (For Mentor Use):
1. (4, 2) | 2. (2, 4) | 3. (3, 2.5) | 4. (1, 2) | 5. (6, 2) | 6. (3, 2) | 7. (1, 4) | 8. (2, 5) | 9. x = (c+d)/(a+b) | 10. Wheat: 5, Barley: 3 | 11. Sheep: 60, Goats: 40 | 15. (3, 4) | 16. (5, 5) | 17. (4, 2) | 18. (2, 7/3) | 19. Dependent (0 = 0) | 20. Inconsistent (0 = 6) | 21. Bread: 6, Honey: 7/3 | 22. Cedar: 4, Oak: 10 | 23. Sheep: 12, Goat: 7 | 24. Red: 4, Blue: 5 | 25. (7, 3)