City Logic Slides CITY LOGIC
SYSTEMS OF EQUATIONS
NC.8.EE.8 | Tier 2 Intervention
WHAT IS A SOLUTION?
The Definition
A solution is an ordered pair (x, y) that makes both equations true at the same time.
The Context
It is the break-even point where two options cost the same or result in the same outcome.
Where lines meet = The Answer
PICK YOUR PLANNING TOOL
Graphing
Best when both are in y = mx + b
Visualizes the "break-even" point clearly
Hard to be exact with fractions or decimals
Substitution
Best when one variable is already isolated
Always gives exact coordinates
More algebra steps (risk of sign errors!)
SCENARIO: DELIVERY FLEET
Technique: Graphing
TASK 1
"A city needs to rent delivery vans. Eco-Van charges $40 plus $1/mile. Speedy-Ship charges $10 plus $3/mile."
Eco-Van: y = x + 40
Speedy-Ship: y = 3x + 10
Discussion
Which van is cheaper for a short 5-mile trip? What about a 30-mile trip?
GRAPH VISUALIZATION (Conceptual)
(15, 55)
Miles (x) Cost (y)
SCENARIO: PARK PERIMETER
Technique: Substitution
Task 2
"A rectangular community park has a perimeter of 400 feet. The length is four times the width."
Eq 1: 2L + 2W = 400
Eq 2: L = 4W
The Substitution Flow
1
Plug 4W into the first equation for L.
2
Solve for W: 2(4W) + 2W = 400
3
Find L by multiplying W by 4.
ERROR DETECTIVE
The Problem:
y = 2x + 5
A student says the solution is (2, -1).
Why are they wrong?
Did they check BOTH equations?
2(2) + 5 = 9
(2) - 3 = -1
Mistake: They found a point that only works for ONE line!
NOW IT'S YOUR TURN
Grab your "City Logic Worksheet." We are going to help the city council decide on some very expensive projects using the math we just reviewed.
Step 1 Identify the Variables
Step 2 Set up Equations
Step 3 Solve & Interpret
City Logic Worksheet CITY LOGIC
Systems of Equations Project
CITIZEN NAME:
DATE:
The Goal
Find the (x, y) coordinates where two lines intersect. In real life, this is the point where two options are exactly equal.
Check Your Work
Plug your x and y back into BOTH original equations. If they don't both work, your lines haven't met yet!
1
THE WATER TOWER CHALLENGE
Two city water towers are being filled. Tower A starts with 200 gallons and fills at 10 gallons per minute. Tower B starts with 50 gallons and fills at 25 gallons per minute.
DEFINE THE EQUATIONS
Tower A:
Tower B:
Solve using substitution or graphing:
INTERPRETATION
After how many minutes will the towers have the same amount? What is that amount?
Minutes:
Gallons:
2
THE COMMUNITY PARK FENCE
A rectangular park has a perimeter of 300 meters. The length (L) is 20 meters longer than the width (W).
SYSTEM OF EQUATIONS
Perimeter:
Comparison:
Solve for L and W using substitution:
Length (L) = ___
Width (W) = ___
Record your final dimensions on the diagram above.
3
SOLAR PANEL SAVINGS
The City Council is comparing two solar installers. Eco-Sun charges $1,000 for installation plus $50/month. Volt-Power has no installation fee ($0) but charges $150/month.
1. Write the system:
Eco-Sun: y =
Volt-Power: y =
2. Find the break-even point:
Decision Matrix
If the city plans to keep the building for 5 years (60 months) , which company will be cheaper overall?
Company:
Explain why using the data from your solving:
THE ERROR DETECTIVE
The Suspect's Work
Solve: y = 2x + 10 and y = 4x + 2
Step 1: 2x + 10 = 4x + 2
Step 2: 10 = 6x + 2
Step 3: 8 = 6x
Step 4: x = 1.33
Mistake happens at Step 2...
Identify the Error
In Step 2, the student moved the 2x to the other side. What did they do wrong?
Correct Solution
Write the correct (x, y) coordinates below:
(____, ____)
City Logic Facilitation Guide FACILITATION GUIDE
City Logic: Systems of Equations
30 MIN | TIER 2 INTERVENTION
Learning Objective
Students will analyze real-world scenarios to set up and solve systems of linear equations using graphing and substitution. Students will interpret the solution (x, y) as a "break-even" point in context.
Standards Alignment
NC.8.EE.8: Analyze and solve pairs of simultaneous linear equations.
Materials Needed
City Logic Slides
City Logic Worksheet
Pencils & Calculators
Straightedges (for graphing)
INSTRUCTIONAL FLOW
05 MIN
The "Break-Even" Concept (Slides 1-3)
Introduce the city theme. Use Slide 2 to anchor the definition of a solution as the unique point where two different plans share the same outcome.
Teacher Prompt:
"If you are choosing between two phone plans, does the 'solution' tell you which one is better, or when they are the same?"
10 MIN
Guided Modeling (Slides 4-5)
Task 1 (Graphing): Emphasize the starting point (y-intercept) and the rate (slope). Physically trace where the lines meet.Task 2 (Substitution): Show how "Length is 4 times Width" creates a direct replacement rule.
15 MIN
The City Planner Worksheet
Students work through Cases 1-3. Circulate and check for common setup errors (e.g., swapping slope and intercept).
Common Misconception
Students often only check their answer in one equation. Insist on checking both.
Tier 2 Support
Provide highlighted "clue words" like 'plus', 'per', or 'is' to help students map words to math symbols.
Wrap Up: The Error Detective
Spend the last 2 minutes on the Error Detective section. This builds metacognition. Ask: "Why would a student add 2x to 4x instead of subtracting it?" (Inverse operations confusion).
City Logic Answer Key ANSWER KEY
CITY LOGIC: SYSTEMS OF EQUATIONS
TEACHER RESOURCE
1
WATER TOWER CHALLENGE
Tower A: y = 10x + 200
Tower B: y = 25x + 50
Substitution Steps:
10x + 200 = 25x + 50
Final Solution:
(10, 300)
2
COMMUNITY PARK FENCE
System: 2L + 2W = 300 and L = W + 20
Solve:
2(W + 20) + 2W = 300W = 65, L = 85
Final Solution:
Length: 85 meters, Width: 65 meters
3
SOLAR PANEL SAVINGS
Eco-Sun: y = 50x + 1000
Volt-Power: y = 150x
Break-Even Point (10, 1500):
At 10 months, both cost $1,500.
Decision Analysis (60 months):
Eco-Sun: 50(60) + 1000 = $4,000Volt-Power: 150(60) = $9,000
!
THE ERROR DETECTIVE
Identify the Error
In Step 2, the student ADDED 2x to 4x (getting 6x) instead of SUBTRACTING 2x from 4x (Inverse operations).
Correct Work
10 = 2x + 2
