The Weaver's Workbook

Edition 11, Lesson 11.1: The First Thread

Part I: The Simple Swap

In these problems, one thread is already isolated. Use substitution to find the Intersection.

1. The Rule of Two:
Thread A: y = 2x
Thread B: x + y = 15
Solution (x, y): ________
2. The Triple Thread:
Thread A: y = 3x - 1
Thread B: 2x + y = 9
Solution (x, y): ________
3. The Inward Search:
Thread A: x = y + 5
Thread B: 3x - y = 19
Solution (x, y): ________
The Weaver's First Check:

Did you remember to find both coordinates? An intersection is not a single number; it is a meeting place of two values.

Part II: Untangling the Threads

In these problems, you must isolate one variable before you can substitute.

4. The Hidden Balance:
Thread A: x + y = 10
Thread B: 2x - y = 2
Solution (x, y): ________
5. The Labor and the Land:
Thread A: y - 2x = 4
Thread B: 3x + 2y = 22
Solution (x, y): ________

Part III: The Witness of the Graph (Geometric Construction)

In Algebra, our logic (Substitution) must match our vision (Graphing). For each problem, briefly sketch the two lines on the provided grid. Mark your estimated intersection, then solve via substitution to find the absolute truth.

6. The Cross-Roads:
Line 1: y = x + 2
Line 2: y = -x + 6
Estimated Intersection: (____, ____)
Exact Solution: (____, ____)

Part IV: Logic and Proof

7. The Law of the Hedge:
Show the algebraic steps for substituting y = 2x - 5 into 4x - 3y = 9. Explain why the parentheses (the hedge) are essential for finding the truth. What happens if the hedge is forgotten?
Step 1: 4x - 3(__________) = 9
Step 2: 4x - (__________ + __________) = 9
Step 3: ...
8. The Proof of Inconsistency:
Solve the following system. If the threads are parallel and never meet, show the mathematical "Lie" that reveals this truth.
Thread 1: y = 3x + 4
Thread 2: 6x - 2y = 10
Conclusion: __________________________________

Part V: Kingdom Modeling

9. The Common Purse:
A community is buying fruit for a feast. Apples (a) cost 2 coins each, and Pears (p) cost 3 coins each. They buy 20 pieces of fruit in total and spend 46 coins.

Rule 1 (Count): a + p = 20
Rule 2 (Cost): 2a + 3p = 46

Find the Intersection to determine how many apples and pears they bought.
Apples: ____ Pears: ____
10. The Master and the Servant:
A Master (M) is 30 years older than his Servant (S). In 10 years, the Master will be twice as old as the Servant.

Rule 1: M = S + 30
Rule 2: M + 10 = 2(S + 10)

How old are they now?
Master: ____ Servant: ____

Part VI: The Weaver's Abstract Thread (Literal Equations)

Sometimes we don't know the numbers, only the relationships. Substitution still works to reveal the form of the truth.

11. The Pure Form:
Thread 1: y = ax + b
Thread 2: y = c

Substitute the value of y from Thread 2 into Thread 1 and solve for 'x'. Your answer will be an expression.
x = ________________
12. The Nested Mystery:
Thread 1: a = b + c
Thread 2: b = 2c

Substitute Thread 2 into Thread 1 to define 'a' entirely in terms of 'c'.
a = ________________

Part VII: Covenant Reflection

13. The Theology of the Intersection:
In your own words, explain why finding the exact intersection point (x, y) is better than just finding one variable. How does this mathematical complete-ness reflect the completeness we find in Christ?
"I vow to show every step of my reasoning, for the path to the Intersection must be clear and true. I will not skip the Weaver's knots, but will tie each one with care. I will honor the Truth as it is revealed, both in the numbers and in the narrative."

Part VIII: The Weaver's Advanced Practice

These problems require careful substitution and may involve fractions or negative numbers.

14. The Mirror Thread:
Thread A: y = x
Thread B: x + y = 10
Solution (x, y): ________
15. The Inverse Weave:
Thread A: x = 2y - 4
Thread B: 3y + x = 11
Solution (x, y): ________
16. The Constant Intersection:
Thread A: 5x + y = 20
Thread B: y = 5
Solution (x, y): ________
17. The Fractional Meeting:
Thread A: y = (1/2)x + 3
Thread B: x + y = 9
Solution (x, y): ________
18. The Negative Path:
Thread A: y = -3x + 10
Thread B: 2x + y = 5
Solution (x, y): ________

Part IX: Special Cases

Not all systems have a single solution. Identify what happens in these cases.

19. The Infinite Thread (Dependent System):
Thread A: x + y = 6
Thread B: 2x + 2y = 12
What happens when you substitute? ________________________________
How many solutions exist? ________________________________
20. The Never-Meeting Thread (Inconsistent System):
Thread A: y = 3x + 1
Thread B: y = 3x - 5
What happens when you substitute? ________________________________
How many solutions exist? ________________________________

Part X: Kingdom Modeling Extended

21. The Two Workers:
Two workers are hired for a job. Worker A earns twice as much per hour as Worker B. Together, they earn $45 per hour.

Rule 1: A = 2B
Rule 2: A + B = 45

Find each worker's hourly wage.
Worker A: $____ per hour | Worker B: $____ per hour
22. The Two Flocks:
A shepherd has sheep (s) and goats (g). There are 3 times as many sheep as goats. The total number of animals is 120.

Rule 1: s = 3g
Rule 2: s + g = 120

How many of each animal does he have?
Sheep: ____ | Goats: ____
23. The Coins in the Chest:
A treasure chest contains gold coins (g) and silver coins (s). The gold coins are worth 5 each, and the silver coins are worth 2 each. There are 50 coins total worth 175.

Rule 1: g + s = 50
Rule 2: 5g + 2s = 175

How many of each coin type?
Gold: ____ | Silver: ____
24. The Journey of Two:
Two travelers start at the same time. Traveler A walks 4 miles per hour. Traveler B rides at 10 miles per hour. They start 36 miles apart and walk toward each other.

Distance covered by A: d₁ = 4t
Distance covered by B: d₂ = 10t
Together: d₁ + d₂ = 36

After how many hours (t) do they meet? How far did each travel?
Time: ____ hours | A traveled: ____ miles | B traveled: ____ miles

Part XI: The Glossary Check

Define each term in your own words.

25. System of Equations:
26. Intersection:
27. Substitution:
28. The Hedge (Parentheses):

Part XII: Reflection Journal

29. The Weaver's Heart:
Read Philippians 2:5-7. Christ "substituted" His divine privileges to take on human form. In what ways is algebraic substitution a shadow of this great truth? How does finding the intersection relate to finding "common ground" with others?
30. My Substitution Story:
Think of a time when you needed to "substitute" something you knew into a confusing situation to bring clarity. How did taking what you already knew and applying it to the unknown help you solve the problem?
The Weaver's Final Check:

Before closing this workbook, verify that:

"I vow to show every step of my reasoning, for the path to the Intersection must be clear and true. I will not skip the Weaver's knots, but will tie each one with care. I will honor the Truth as it is revealed, both in the numbers and in the narrative. I am ready for the Gardener's lesson."
Answer Key Reference (For Mentor Use):
1. (5, 10) | 2. (2, 5) | 3. (7, 2) | 4. (4, 6) | 5. (2, 8) | 6. (2, 4) | 8. Inconsistent (No solution) | 9. Apples: 14, Pears: 6 | 10. Master: 50, Servant: 20 | 11. x = (c-b)/a | 12. a = 3c | 14. (5, 5) | 15. (2, 3) | 16. (3, 5) | 17. (4, 5) | 18. (5, -5) | 19. Dependent (0 = 0, infinite solutions) | 20. Inconsistent (6 = -4, no solution) | 21. A: $30, B: $15 | 22. Sheep: 90, Goats: 30 | 23. Gold: 25, Silver: 25 | 24. t = 2.57 hours approx