Parallel lines NEVER meet They share the same slope!
PERPENDICULAR = FLIP & FLIP SIGN
If m = 2, perp. m = −½ (Negative reciprocal)
VERTICAL → UNDEFINED
x = 3 is vertical Slope = undefined (÷ by 0!)
HORIZONTAL → ZERO
y = 5 is horizontal Slope = 0 (flat line)
📐 Key Formulas
Slope-Intercept
y = mx + b
Point-Slope
y − y₁ = m(x − x₁)
Standard Form
Ax + By = C
Slope Formula
m = (y₂ − y₁)/(x₂ − x₁)
Slope-Intercept Form
Which equation is written in slope-intercept form?
Remember: slope-intercept form looks like y = mx + b
💡 Quick Exampley = 3x + 5 → slope = 3, y-intercept = 5 ✓ This IS slope-intercept form. 2x + y = 7 → This is Standard Form, NOT slope-intercept.
⚠️ Trick Alert: The equation must be solved FOR y (y alone on left side!)
📖 Explanation
Slope-intercept form = y = mx + b → y must be alone on the left.
❌ A: 3x + 2y = 6 → Standard Form
✅ B: y = −2x + 7 → y is alone! m = −2, b = 7 ✓
❌ C: x = 4y − 1 → x is isolated, not y
❌ D: 5x − y = 0 → Standard Form
Finding Slope
What is the slope of the line passing through
points (1, 3) and (5, 11)?
💡 Quick Example
Points (0, 2) and (3, 8) → m = (8−2)/(3−0) = 6/3 = 2
Always: Top = y-change, Bottom = x-change. (RISE over RUN!)
⚠️ Trick Alert: Keep the ORDER consistent! Don't mix up which point is "1" and "2"
📖 Explanation
Use slope formula: m = (y₂ − y₁) / (x₂ − x₁)
Points: (1, 3) and (5, 11)
m = (11 − 3) / (5 − 1) = 8 / 4 = 2
✅ Answer: m = 2
The line goes up 2 units for every 1 unit right → positive slope ↗
Y-Intercept
For the equation y = 4x − 9,
what is the y-intercept?
💡 Quick Example
y = 7x + 3 → y-intercept = 3 (it's always the "b" in y = mx + b)
y = −x − 6 → y-intercept = −6
📌 Memory: b = the point where the line CROSSES the y-axis (when x = 0)
📖 Explanation
In y = mx + b:
→ m is the slope, b is the y-intercept
y = 4x − 9 → b = −9
Verify: plug in x = 0 → y = 4(0) − 9 = −9 ✓
❗ Don't pick "4" — that's the slope, not the intercept!
Vertical Lines
What is the slope of the vertical line x = −3?
💡 Quick Example
Vertical line x = 7 → slope = UNDEFINED
Horizontal line y = 7 → slope = 0
(Think: a wall is vertical and you can't climb an undefined slope!)
⚠️ Most Common Mistake: Students write "0" for vertical lines. WRONG! 0 is for HORIZONTAL!
📖 Explanation
Vertical line: x = constant → run = 0
m = rise / run = anything / 0 = UNDEFINED!
Division by zero is undefined in math!
✅ Vertical line → Undefined slope
✅ Horizontal line → Slope = 0
🧠 Memory trick: "Vertical = Void (undefined)"
Parallel Lines
Line ℓ has equation y = 3x + 1.
Which line is parallel to line ℓ?
💡 Quick Example
y = 2x + 5 is parallel to y = 2x − 8 (same slope = 2, different b)
PARALLEL = Same slope, different y-intercept
📌 PARALLEL = SAME SLOPE. The y-intercept MUST be different (same = identical line!)
📖 Explanation
Original line: y = 3x + 1 → slope m = 3
For parallel: same slope (m = 3), different b
❌ A: slope = −⅓ (perpendicular, not parallel!)
✅ B: y = 3x − 5 → slope = 3 ✓ different b ✓
❌ C: same slope AND same b → it's the IDENTICAL line!
❌ D: slope = ⅓ (not the same)
🧠 Parallel rails on a train track → same direction, never meet!
Perpendicular Lines
Line ℓ has slope m = 4.
What is the slope of a line perpendicular to ℓ?
💡 Quick Example
m = 2 → perpendicular slope = −½
m = −⅓ → perpendicular slope = 3
Step: FLIP the fraction, then FLIP the sign
❌ B: That's the same slope (parallel!)
❌ C: Forgot the negative sign!
❌ D: Forgot to flip the fraction (just changed sign)
Point-Slope Form
Write the equation of the line with slope m = −2
passing through point (3, 5).
Which point-slope equation is correct?
💡 Quick Example
m = 3, point (2, 1): y − 1 = 3(x − 2)
Formula: y − y₁ = m(x − x₁) → plug in point for x₁ and y₁
⚠️ Trick Alert: Signs flip inside! Point (3, 5) → write (x − 3) NOT (x + 3)!
📖 Explanation
Formula: y − y₁ = m(x − x₁)
m = −2, point (3, 5) → x₁ = 3, y₁ = 5
y − 5 = −2(x − 3)
❌ A: x + 3 should be x − 3 (sign error!)
❌ B: y + 5 should be y − 5
✅ C: Correct! Both signs are right ✓
❌ D: That's not even point-slope form
X-Intercept
What is the x-intercept of the line 2x + 3y = 12?
💡 Quick Example
x-intercept of y = 2x − 6: set y = 0 → 0 = 2x − 6 → x = 3
x-intercept = (3, 0) ← always has y = 0!
📌 X-intercept: set y = 0 and solve for x. Y-intercept: set x = 0 and solve for y.
📖 Explanation
X-intercept → set y = 0 and solve:
2x + 3(0) = 12 → 2x = 12 → x = 6
✅ x-intercept = (6, 0)
❌ A: That's the y-intercept! (set x = 0: 3y = 12, y = 4... wait, check: 3y=12, y=4. A is wrong too)
❌ C: 12 is the constant, not the intercept
Common mistake: forgetting to divide by the coefficient of x!
Reading a Graph
A line passes through (0, −2) and (4, 6).
What is the equation of this line?
💡 Strategy
Step 1: Find slope m = (6−(−2))/(4−0) = 8/4 = 2
Step 2: The point (0, −2) is the y-intercept → b = −2
📖 Explanation
Slope: m = (6 − (−2)) / (4 − 0) = 8/4 = 2
(0, −2) is on y-axis → y-intercept b = −2
y = 2x − 2
✅ Check with (4, 6): y = 2(4) − 2 = 6 ✓
❌ A: b should be −2, not +2
❌ D: slope ½ is wrong (don't flip rise and run!)
Standard Form → Slope-Intercept
Convert 6x − 2y = 10 to slope-intercept form.
What are the slope and y-intercept?
💡 Quick Example
4x − 2y = 8 → isolate y:
−2y = −4x + 8 → divide by −2 → y = 2x − 4
⚡ Key: When you divide by a NEGATIVE, ALL signs flip!
⚠️ Most Common Mistake: Forgetting to flip signs when dividing by negative number!
📖 Explanation
Solve for y step by step:
6x − 2y = 10
Subtract 6x: −2y = −6x + 10
Divide by −2: y = 3x − 5
✅ m = 3, b = −5
❌ D: b = −5 not +5 (forgot to flip sign when dividing by −2!)
❌ A: Those are the raw coefficients — you can't just read them off!