Linear Mastery Review Slides
GRADE 8 MATHEMATICS • TOPIC 2
COMPREHENSIVE EXAM REVIEW
Full Assessment Study Guide Review
Linear Relations & Equations
Master slopes, rates of change, multi-step algebraic solutions, and inequalities with step-by-step example breakdowns.
Multi-Step Algebra
Fractions, decimals & signs
Slope & Graphs
Unit rate, \(y = mx + b\)
Solutions & Rules
\(0, 1, \infty\) and inequalities
SKILL 1
Equations with Decimals & Fractions
Study Guide Q1, Q4 & Q7
Core Strategies to Clear Obstacles
- • Fractions: Multiply every single term by the LCD to eliminate denominators completely.
- • Decimals: Distribute carefully. Watch negatives: \(-0.2(-30) = +6\).
- • Isolate: Collect all variable terms on one side and constants on the other.
Pro Tip: To divide \(\frac{32}{0.8}\), scale by 10 to get \(\frac{320}{8} = 40\).
Example: Clear Fractions with LCD 12
\(\frac{3}{4}x - \frac{1}{3}x = x - 1\)
1. Multiply by 12: \(9x - 4x = 12x - 12\)
2. Combine terms: \(5x = 12x - 12\)
3. Subtract \(12x\): \(-7x = -12\)
Final Solution: \(x = \frac{12}{7} = 1\frac{5}{7}\)
Always distribute to every term inside parentheses before moving variables across the equal sign.
SKILL 2
Proportional Relationships & Negative Slope
Study Guide Q2
Submarine Descent Analysis
Height Above Sea Level vs. Time
0 2 4 6s 0 -8 -16 -24ft
Points: \((0,0)\), \((2,-8)\), \((4,-16)\), \((6,-24)\)
Evaluating Key Statements
✓ Proportional: Passes straight through origin \((0,0)\).
✓ Negative Slope: Slants downward from left to right.
✗ Slope value: \(m = \frac{-8 - 0}{2 - 0} = -4\) (not \(-8\)).
✓ Equation: \(y = -4x\). (y-intercept is \(0\), not \(-4\)).
Key Rule: A relation is proportional if and only if it is linear and passes directly through \((0,0)\).
Descent Rate: \(4\text{ ft/s downward}\) Constant of Proportionality \(k = m = -4\)
SKILL 3
Slope as Unit Rate in Real Contexts
Study Guide Q3
Water in Sink Graph
Volume (gallons) vs. Time (minutes)
0 3 6 9m 0 1 2 3 4g
Slope \(m = \frac{\Delta y}{\Delta x} = \frac{1\text{ gallon}}{3\text{ minutes}} = \frac{1}{3}\text{ gal/min}\)
Common Trap: Inverting the Ratio!
Do not calculate \(\frac{\text{run}}{\text{rise}} = \frac{3}{1} = 3\). Slope is always vertical change over horizontal change!
Incorrect: 3 gallons per minute
Correct: \(\frac{1}{3}\) gallon per minute
Answering the Evaluation Prompt
"Maya is not correct because the rate of water flowing is equal to the slope, which is less than 3 gal/min."
Always include units: \(\frac{\text{vertical units (gallons)}}{\text{horizontal units (minutes)}}\).
SKILL 4
Slope-Intercept Form: \(y = mx + b\)
Study Guide Q5 & Q9
Anatomy of Real-World Linear Models
\(y = \) \(m\)\(x + \) \(b\)
m
Slope / Rate of Change:
Variable rate ("per hour", "per mile").
b
y-Intercept / Initial Value:
One-time flat fee or starting value at \(x = 0\).
Graph checkpoint: Intersects the y-axis at \((0, b)\), not \((0,0)\)!
Paddleboard Rental Modeling
Shop charges $15 flat fee plus $8.50 per hour. Total cost paid is $66.
1. Model equation: \(8.50h + 15 = 66\)
2. Subtract flat fee ($15): \(8.50h = 51\)
3. Divide by rate ($8.50): \(h = \frac{51}{8.50}\)
Rental Duration: \(h = 6\text{ hours}\)
Parking Garage ($4 flat + $3/hr): \(y = 3x + 4\) Starts at \((0,4)\) with slope \(+3\)
SKILL 5
Number of Solutions to Linear Equations
Study Guide Q6
Case 1
One Solution
Different Slopes (\(a \neq c\))
Variables do NOT cancel out. Lines intersect at one point \((x, y)\).
\(3(2x - 2) = -6x - 6\)
\(6x - 6 = -6x - 6\)
\(12x = 0 \rightarrow x = 0\)
Case 2
No Solution
Same Slope, Diff Constants
Variables cancel out leaving a FALSE statement. Lines are parallel.
\(-3(2x - 2) = -6x - 4\)
\(-6x + 6 = -6x - 4\)
\(6 = -4\) (FALSE!)
Case 3
Infinitely Many
Identical Expressions
Both sides simplify to the exact same expression. Leaves a TRUE identity.
\(3(2x - 2) = 6x - 6\)
\(6x - 6 = 6x - 6\)
\(-6 = -6\) (TRUE!)
Quick Test Hack: Compare simplified \(x\)-coefficients first! \(a \neq c \rightarrow\) Exactly 1 solution immediately.
SKILL 6
Comparing Rates: Graph vs. Equation
Study Guide Q8
Elena's Travel (Graph)
Visual
0 120 240mi 0 2 4h
Unit rate from point \((2, 240)\):
Rate = \(\frac{240\text{ miles}}{2\text{ hours}} = 120\text{ mph}\)
Marcus's Travel (Equation)
Algebraic
Given Linear Equation:
\(y = 130x\)
In \(y = mx\), the coefficient \(m\) is the unit rate!
Unit rate from coefficient \(m\):
Rate = \(130\text{ mph}\)
VS
Elena: 120 mph | Marcus: 130 mph
Marcus traveled at a faster speed (\(130 > 120\))
SKILL 7
Solving Multi-Step Inequalities
Study Guide Q10
Walkthrough: Study Guide Problem
\(2(k + 4) - 6 < 8\)
1. Distribute 2: \(2k + 8 - 6 < 8\)
2. Combine constants: \(2k + 2 < 8\)
3. Subtract 2: \(2k < 6\)
4. Divide by +2: \(k < 3\)
Final Solution: \(k < 3\)
The Negative Sign Flip Rule
Solve inequalities just like equations with one crucial difference:
When you multiply or divide by a negative number, you must REVERSE / FLIP the inequality sign!
Dividing by positive (+2): Sign stays \(<\)
Dividing by negative (-2): Sign flips to \(>\)
Watch Out: Subtracting a number does NOT flip the sign. Only negative multiplication or division flips it!
Inequality Graphs: \(<\) open circle left | \(>\) open circle right Math 8 Topic 2
SUMMARY
Topic 2 Formula Card & Test Day Checklist
Ready for 100%!
Slope & Rate of Change
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}\)
Positive = slants up right | Negative = slants down right
Linear Models
\(y = mx\) (proportional, passes \((0,0)\))
\(y = mx + b\) (starts at \((0,b)\), \(m = \text{rate}\))
Number of Solutions
- • Diff. slopes (\(a \neq c\)) \(\rightarrow\) 1 Solution
- • Same slope, diff constant \(\rightarrow\) No Solution
- • Identical both sides \(\rightarrow\) Infinitely Many
Execution Reminders
- • Clear fractions: multiply every term by LCD.
- • Flip inequality ONLY when \(\times\) or \(\div\) by negative.
- • Double check signs during distribution!
You are fully prepared for every question on the Topic 2 Assessment!
