Linear Codebreakers Worksheet
Linear Codebreakers
8th Grade Mathematics • Two-Variable Linear Equations
Name:
Date: Period:
Field Formula Reference:
Slope-Intercept: \( y = mx + b \)
Slope (\(m\)): \( \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x} \)
Solution: Makes statement true
MISSION 1
Solution Verifier: Test the Coordinates
Substitute the given point \( (x, y) \) into each equation. Show your substitution work, then check if it forms a true statement.
1. \( y = 3x - 5 \) Point: (4, 7)
Is (4, 7) a solution?
YES NO
2. \( 2x + y = 10 \) Point: (3, 4)
Is (3, 4) a solution?
YES NO
3. \( y = -\frac{1}{2}x + 2 \) Point: (6, -2)
Is (6, -2) a solution?
YES NO
MISSION 2
Decode the Rule: From Table to Linear Equation
Find the rate of change (slope \(m\)) and initial value (y-intercept \(b\)), then formulate the equation in \( y = mx + b \).
Data Set A
| \(x\) |
\(y\) |
| -2 |
-7 |
| 0 |
-1 |
| 2 |
5 |
| 4 |
11 |
Slope (\(m = \frac{\Delta y}{\Delta x}\)):
y-intercept (\(b\)):
Linear Equation:
\(y = \)
Data Set B
| \(x\) |
\(y\) |
| -3 |
10 |
| 0 |
4 |
| 3 |
-2 |
| 6 |
-8 |
Slope (\(m = \frac{\Delta y}{\Delta x}\)):
y-intercept (\(b\)):
Linear Equation:
\(y = \)
MISSION 3
Graphing Coordinates & Linear Trajectories
Identify the slope and y-intercept, calculate three solution points, then plot and draw the straight line with arrows.
Line 1: \( y = 2x - 3 \)
Slope (\(m\)):
y-int (\(b\)):
Table of Values:
x y
Line 2: \( y = -\frac{1}{2}x + 2 \)
Slope (\(m\)):
y-int (\(b\)):
Table of Values:
x y
8th Grade Linear Relationships • Standards: CCSS.MATH.CONTENT.8.EE.B.5 & 8.EE.B.6 Page 1 of 2
Linear Codebreakers • Part II
Form Conversion, Real-World Modeling & Error Analysis
Name:
MISSION 4
Standard to Slope-Intercept Form
Isolate variable \( y \) using inverse operations to rewrite each equation into \( y = mx + b \) form, then state the slope and y-intercept.
1. \( 4x + y = 9 \)
Show your algebraic steps:
Equation: \( y = \)
\(m = \) \(b = \)
2. \( 3x - 2y = 8 \)
Show your algebraic steps:
Equation: \( y = \)
\(m = \) \(b = \)
3. \( -6x + 3y = -12 \)
Show your algebraic steps:
Equation: \( y = \)
\(m = \) \(b = \)
MISSION 5
Real-World Linear Modeling & Interpretation
Translate practical scenarios into two-variable linear equations and use them to solve problems.
Scenario A: The Apex Drone Rental
Apex Aerial charges a flat reservation insurance fee of $15 plus $20 per hour of drone flight time. Let \( x \) represent the number of hours rented and \( y \) represent the total cost in dollars.
a. Write a linear equation modeling total cost \( y \):
\( y = \)
b. Identify what each component represents:
Slope (\(m = 20\)):
y-int (\(b = 15\)):
c. Calculate total cost for renting the drone for 4.5 hours:
Scenario B: Deep Sea Submersible Descent
A research submersible starts exploring at a depth of -50 meters (50m below surface) and steadily descends at a constant rate of 15 meters per minute (\( -15 \) m/min). Let \( x \) be time in minutes and \( y \) be elevation in meters.
a. Write a linear equation modeling depth \( y \):
\( y = \)
b. What will the submarine's depth be after 12 minutes?
c. If the target trench is at -290 meters, how many minutes will it take to reach?
DETECTIVE LAB
Error Analysis: Spot the Codebreaker's Glitch
Agent Taylor was tasked with finding the equation of a line passing through the points \((2, 7)\) and \((5, 16)\). Here is Taylor's calculation:
Step 1: \( m = \frac{16 - 7}{5 - 2} = \frac{9}{3} = 3 \)
Step 2: \( y = 3x + b \)
Step 3: \( 2 = 3(7) + b \implies 2 = 21 + b \)
Step 4: \( b = 2 - 21 = -19 \)
Result: \( y = 3x - 19 \)
1. Explain Taylor's specific error:
2. Provide the corrected work and correct equation:
Correct Equation: \( y = \)
8th Grade Linear Relationships • Standards: CCSS.MATH.CONTENT.8.EE.B.5 & 8.EE.B.6 Page 2 of 2
