The Watchman's Journal

Edition 11, Lesson 11.3: Graphing Intersections

Part I: The Single Intersection

For each problem, graph both lines carefully using a ruler. Identify the Intersection and verify it algebraically.

1. The Paths of Peace:
Line 1: y = x + 1
Line 2: y = -x + 5
Intersection (x, y): (____, ____)
2. The Ascending Witness:
Line 1: y = 2x - 4
Line 2: y = 1/2x + 2
Intersection (x, y): (____, ____)

Part II: Parallel and Identical Stories

Graph these systems and identify if they are Unique, Inconsistent (Parallel), or Dependent (Identical).

3. The Separate Ways:
Line 1: y = 2x + 3
Line 2: y = 2x - 1
Story Type: __________________ Solution: __________________
4. The Harmonious Song:
Line 1: y = -x + 4
Line 2: 2x + 2y = 8
Story Type: __________________ Solution: __________________

Part III: Untangling the Witness

Convert these Standard Form rules into y=mx+b form before graphing.

5. The Market Meeting:
Rule 1: x + y = 6
Rule 2: 3x - y = 2
Rule 1 (y=mx+b): ________________
Rule 2 (y=mx+b): ________________
Intersection (x, y): (____, ____)
6. The Hidden Slope:
Rule 1: 2x + y = 8
Rule 2: x - 2y = 4
Rule 1 (y=mx+b): ________________
Rule 2 (y=mx+b): ________________
Intersection (x, y): (____, ____)

Part IV: The Inverse Witness (Discovery)

Sometimes we see the intersection and one path, and we must discover the other.

7. The Missing Messenger:
We know the two messengers met at (3, 6). Messenger A followed the rule y = 2x. Messenger B followed a rule with a slope of -1.

Task: Graph the meeting point and Messenger A's line. Draw Messenger B's line through the meeting point using the slope of -1. What is Messenger B's y-intercept (b)?
Messenger B's Rule: y = -x + ____

Part V: Kingdom Modeling

8. The Messenger's Meeting:
Messenger A starts at y=10 and moves at a slope of -1 (y = -x + 10).
Messenger B starts at y=0 and moves at a slope of 1 (y = x).

Graph their journey. At what coordinate do they exchange their messages?
Meeting Point: (____, ____)
9. The Stewardship of Time:
A student has 10 hours for study and rest. Let x be study hours and y be rest hours. (x + y = 10).
He wants to spend 2 more hours studying than resting (x = y + 2).

Graph these two rules of life. Where is the Intersection of his stewardship?
Study Hours: ____ Rest Hours: ____

Part VI: The Watchman's Reflection

10. The Witness of the Grid:
Why is it important to use a ruler and a sharpened pencil when graphing a system? What happens to the "Truth" if the line drifts even slightly? How does this reflect our walk with the Lord?
11. Inconsistent vs. Dependent:
In your own words, describe the difference between a system that has *no solution* and a system that has *infinite solutions*. Use the metaphor of two people talking to explain. How can we tell them apart just by looking at their Slopes (m) and Intercepts (b)?
"I vow to be a faithful witness. I will draw my lines with precision and verify my vision with logic. I will not accept a blurry intersection, but will seek the sharp and clear truth of the coordinate."

Part VII: Advanced Graphing Practice

Graph these systems and identify the intersection. Convert to slope-intercept form if needed.

12. The Steeper Path:
Line 1: y = 3x - 4
Line 2: y = -x + 8
Intersection (x, y): (____, ____)
13. The Gradual Slope:
Line 1: y = (1/3)x + 1
Line 2: y = -2x + 8
Intersection (x, y): (____, ____)
14. The Standard Form Challenge:
Rule 1: 3x + y = 9
Rule 2: x - y = -1
Rule 1 (y=mx+b): ________________
Rule 2 (y=mx+b): ________________
Intersection (x, y): (____, ____)
15. The Perpendicular Meeting:
Line 1: y = 2x + 1
Line 2: y = -1/2x + 6
Intersection (x, y): (____, ____)

Part VIII: System Classification

For each system, graph the lines and classify as Consistent/Independent, Inconsistent, or Dependent.

16. System A:
Line 1: y = x + 3
Line 2: y = x - 2
Classification: ________________
Number of Solutions: ________________
How can you tell from the slopes and intercepts? ___________________________
17. System B:
Line 1: y = 2x + 4
Line 2: 2y = 4x + 8
Classification: ________________
Number of Solutions: ________________
What happened when you graphed the second line? ___________________________

Part IX: Kingdom Modeling with Graphs

18. The Race to the Gate:
Runner A starts at position 0 and runs at 5 units per minute: y = 5x
Runner B starts at position 20 and runs at 3 units per minute: y = 3x + 20

Question: Do they ever meet? Graph both lines to find out.
Meeting Point (if any): (____, ____)
What does the graph tell you about their speeds? ___________________________
19. The Converging Paths:
Traveler A starts at y = 12 and descends at 2 units per hour: y = -2x + 12
Traveler B starts at y = 0 and ascends at 1 unit per hour: y = x

Graph their journeys. At what time and position do they meet?
Time of Meeting: x = ____
Position of Meeting: y = ____
20. The Water Tanks:
Tank A has 100 gallons and drains at 10 gallons per hour: y = -10x + 100
Tank B has 20 gallons and fills at 5 gallons per hour: y = 5x + 20

Graph both tanks. When will they have the same amount of water? How much?
Time: x = ____ hours
Amount in each tank: y = ____ gallons

Part X: The Three Methods Comparison

21. The Master's Challenge:
Solve this system using ALL THREE methods, then compare.

Thread A: y = x + 2
Thread B: 2x + y = 11

Method 1: Graphing

Estimated Intersection: (____, ____)

Method 2: Substitution

Exact Solution: (____, ____)

Method 3: Elimination

Exact Solution: (____, ____)

Reflection: Did all three methods give you the same answer? Which method was fastest for this problem? Which gave you the most confidence in your answer?

Part XI: The Glossary Check

Define each term in your own words.

22. Coordinate Plane:
23. Slope (m):
24. Y-Intercept (b):
25. Visual Witness:

Part XII: Reflection Journal

26. The Watchman's Heart:
Read Deuteronomy 19:15: "A matter must be established by the testimony of two or three witnesses."

How does graphing serve as a "visual witness" to confirm what you calculated algebraically? Why is it important to have multiple witnesses for truth—both in math and in life?
27. My Watchman Story:
Think of a time when you could "see the big picture" that others couldn't. Maybe you noticed where two things were heading, or you saw how a situation would end before it happened. How is this like being a mathematical watchman?
28. Edition 11 Summary:
You have now learned three methods: Substitution (The Weaver), Elimination (The Gardener), and Graphing (The Watchman). In your own words, when would you use each method?

I would use Substitution when:
I would use Elimination when:
I would use Graphing when:
The Watchman's Final Check:

Before closing this workbook, verify that:

"I vow to be a faithful witness. I will draw my lines with precision and verify my vision with logic. I will not accept a blurry intersection, but will seek the sharp and clear truth of the coordinate. I am ready for Edition 12: The Root."
Answer Key Reference (For Mentor Use):
1. (2, 3) | 2. (4, 0) | 3. Parallel (No solution) | 4. Dependent (Infinite solutions) | 5. (2, 4) | 6. (4, 2) | 7. (3, 6) | 8. (5, 5) | 9. Study: 6, Rest: 4 | 12. (3, 5) | 13. (3, 2) | 14. (2, 3) | 15. (2, 5) | 16. Inconsistent (Parallel, no solution) | 17. Dependent (Same line, infinite solutions) | 18. They never meet (same slope: 5 ≠ 3+20/x, parallel if starting positions considered) OR recheck problem setup | 19. (4, 4) | 20. (16/3 ≈ 5.33, 46.67 gallons) | 21. (3, 5)