Linear Relations Cheat Sheet
\(m\)
Grade 8 Math Topic 2 Quick Study Guide
Linear Relations & Equations Cheat Sheet
Slope • Systems • Modeling • Inequalities
Exam Reference Card
1. Slope & Rate of Change
\(m = \frac{\Delta y}{\Delta x}\)
\[ m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \text{Unit Rate} \]
Always put vertical change (\(y\)) over horizontal change (\(x\))
↗ Positive
Uphill L \(\to\) R
↘ Negative
Downhill L \(\to\) R
→ Zero (\(0\))
Flat horizontal
↑ Undefined
Vertical line
• Proportional: Passes through origin \((0,0)\). Equation is \(y = mx\). Constant ratio \(\frac{y}{x} = m\).
• Non-Proportional: Does NOT pass through \((0,0)\). Starts at \(y\)-intercept \((0, b)\).
2. Slope-Intercept Form: \(y = mx + b\)
\(m\) = Rate of Change
• "Per", "each", "every", "hourly rate"
• Controls the steepness / direction
\(b\) = Initial Value
• "Flat fee", "deposit", "start value"
• Graph crosses \(y\)-axis at \((0, b)\)
Real-World Word Problem Template:
"A rental shop charges a $15 flat fee plus $8.50 per hour. Total paid is $66."
Equation: \(8.50h + 15 = 66\) \(\to\) \(8.50h = 51\) \(\to\) \(h = 6\text{ hrs}\)
3. Comparing Rates Across Formats
Convert all representations into their unit rate (slope \(m\)):
| Format | How to find Unit Rate (\(m\)) | Example |
|---|
| Graph | Pick a clear point \((x,y)\); calculate \(\frac{y}{x}\) | \((2, 240) \to \frac{240}{2} = 120\text{ mph}\) |
| Equation | Identify coefficient of \(x\) in \(y = mx\) | \(y = 130x \to \text{rate} = 130\text{ mph}\) |
| Compare: \(130 > 120\), so the equation train is faster by 10 mph. | | |
4. Number of Solutions Checklist
High-Yield
Simplify both sides to \(ax + b = cx + d\) and inspect:
Exactly ONE Solution \(a \neq c\)
Rule: Variable coefficients are DIFFERENT.
\(6x - 6 = -6x - 6 \implies 12x = 0 \implies x = 0\)
NO Solution \(a = c\), \(b \neq d\)
Rule: Same \(x\) terms, DIFFERENT constants (FALSE statement!).
\(-6x + 6 = -6x - 4 \implies 6 = -4\) (False!)
INFINITELY Many Solutions \(a = c\), \(b = d\)
Rule: Both sides simplify to the exact same expression.
\(6x - 6 = 6x - 6 \implies -6 = -6\) (Identity/True!)
5. Solving Equations: Fractions & Decimals
Strategy A: Clear Fractions with LCD Multiply all terms
Solve: \(\frac{3}{4}x - \frac{1}{3}x = x - 1\) (LCD of 4 and 3 is 12):
\(12(\frac{3}{4}x) - 12(\frac{1}{3}x) = 12(x - 1)\)
\(\implies 9x - 4x = 12x - 12\)
\(\implies 5x = 12x - 12 \implies -7x = -12 \implies x = \frac{12}{7} = 1\frac{5}{7}\)
Strategy B: Careful Distribution with Signs
Watch negatives! \(-0.2(x - 30) = -0.2x + 6\) (negative \(\times\) negative = positive!)
\(-0.2x + 6 = 38 - x \implies 0.8x = 32 \implies x = \frac{32}{0.8} = 40\)
6. Inequalities & The "Sign Flip" Rule
⚠️ THE GOLDEN RULE:
Flip the inequality symbol (\(< \leftrightarrow >\) and \(\le \leftrightarrow \ge\)) ONLY when multiplying or dividing by a negative number!
Step-by-Step Model:
\(2(k + 4) - 6 < 8\)
\(2k + 8 - 6 < 8\)
\(2k + 2 < 8\)
\(2k < 6 \implies \mathbf{k < 3}\)
Number Line Guide:
• \(<\) or \(>\): Open Circle \(\circ\)
• \(\le\) or \(\ge\): Closed Circle \(\bullet\)
• \(<\) or \(\le\): Shade Left \(\leftarrow\)
• \(>\) or \(\ge\): Shade Right \(\rightarrow\)
Top 3 Test Traps
- 1. Forgetting \(\frac{\text{rise}}{\text{run}}\) puts \(y\) on top, not \(x\)
- 2. Multiplying by LCD must reach every term (including constants)
- 3. Only flip inequality signs for negative multipliers/divisors
Math 8 • Topic 2
