🎯 Learning Objectives
- Slope-intercept, point-slope, and standard forms: when to use each.
- How to read slope and y-intercept off a graph.
- How to find slope from any two points without making sign errors.
- What parallel and perpendicular lines look like algebraically.
- Strategies for matching equations to graphs.
- Practice questions with worked solutions.
1. Three Forms of a Line
Slope-Intercept Form
y = m·x + b
Point-Slope Form
y − y₁ = m·(x − x₁)
Standard Form
A·x + B·y = C
Form | When to Use |
Slope-intercept (y = mx + b) | You know slope and y-intercept; or you want to graph fast. |
Point-slope | You know slope and ANY single point on the line. |
Standard (Ax + By = C) | You're given an equation in this form (often word problems). |
2. Slope: The 'Steepness' of a Line
Slope between two points
m = (y₂ − y₁) / (x₂ − x₁)
Slope is rise over run. Positive slope rises left-to-right; negative slope falls; zero slope is horizontal; undefined slope is vertical.
Slope | Looks Like | Example |
Positive (m > 0) | Rising line ↗ | y = 2x + 1 |
Negative (m < 0) | Falling line ↘ | y = −3x + 4 |
Zero (m = 0) | Horizontal line : | y = 5 |
Undefined | Vertical line | | x = 3 |
3. Reading a Linear Graph
STEP 1: Find the y-intercept (where the line crosses the y-axis). That's b.
STEP 2: Pick any two clean lattice points. Compute rise/run. That's m.
STEP 3: Write the equation: y = m·x + b
4. Parallel and Perpendicular Lines
❓ Comparing two lines: what relationship? | ||
Same slope, different y-int ▼ | Slopes multiply to −1 ▼ | |
PARALLEL (never meet) | PERPENDICULAR (90° angle) | |
💡 Negative Reciprocal: How to Find a Perpendicular Slope
Flip the fraction and change the sign. Slope of 2/3 → perpendicular slope is −3/2. Slope of −5 → perpendicular slope is 1/5.
📝 Example: Find the equation of a line through (4, 1) parallel to y = 3x − 7.
Step 1: Parallel → same slope. m = 3.
Step 2: Use point-slope: y − 1 = 3(x − 4)
Step 3: Simplify: y = 3x − 11
5. Strategies
⚡ STRATEGY 1: Quick check for matching equations
When matching an equation to a graph, plug x = 0 into the equation. The result is the y-intercept: see if it matches the graph.
⚡ STRATEGY 2: Slope sign test
Read the graph from left to right. A rising line has positive slope; a falling line has negative slope. Use two points when a numerical slope is needed.
⚡ STRATEGY 3: Convert standard form fast
To get slope from Ax + By = C, just compute m = −A/B. Don't waste time solving for y.
⚠️ Common Mistakes
- Subtracting coordinates in the wrong order: always (y₂ − y₁) over (x₂ − x₁): same order!
- Confusing parallel (same slope) with perpendicular (negative reciprocal slope).
- Mixing up rise and run: rise is vertical (y), run is horizontal (x).
- Forgetting that horizontal lines have slope 0, not undefined.
6. Summary
📌 Key Takeaways
- Slope-intercept: y = m·x + b. m is slope, b is y-intercept.
- Slope formula: m = (y₂ − y₁) / (x₂ − x₁)
- Parallel: Same slope, different intercept.
- Perpendicular: Slopes are negative reciprocals (product = −1).
- Horizontal line: y = constant; slope = 0.
- Vertical line: x = constant; slope is undefined.
7. Practice: 10 Questions
Q1. What is the slope of the line passing through (2, 3) and (6, 11)?
A) 1 B) 2 C) 3 D) 4
Hint: (11 − 3)/(6 − 2) = 8/4 = 2.
Q2. A line has equation y = −4x + 7. What is its y-intercept?
A) −4 B) 4 C) −7 D) 7
Hint: b is the constant. y-int is 7.
Q3. What is the slope of the line 3x + 2y = 12?
A) 3/2 B) −3/2 C) 2/3 D) −2/3
Hint: Use −A/B = −3/2.
Q4. Which line is parallel to y = (1/2)x + 3?
A) y = 2x + 3 B) y = −(1/2)x + 1 C) y = (1/2)x − 5 D) y = −2x + 7
Hint: Parallel = same slope (1/2).
Q5. Which line is perpendicular to y = 5x − 1?
A) y = 5x + 7 B) y = −5x + 2 C) y = (1/5)x − 1 D) y = −(1/5)x + 4
Hint: Negative reciprocal of 5 is −1/5.
Q6. A line passes through (0, −3) with slope 4. What is its equation?
A) y = 4x − 3 B) y = −3x + 4 C) y = 4x + 3 D) y = −4x − 3
Hint: y-int = −3, slope = 4: y = 4x − 3.
Q7. The line y = mx + 2 passes through (3, 8). What is m?
A) 1 B) 2 C) 3 D) 6
Hint: Plug in: 8 = 3m + 2 → m = 2.
Q8. A horizontal line passes through (5, −2). What is its equation?
A) x = 5 B) x = −2 C) y = 5 D) y = −2
Hint: Horizontal: y = constant = −2.
Q9. Two points: (−1, 4) and (3, −4). What is the slope?
A) −2 B) 2 C) −1/2 D) 1/2
Hint: (−4 − 4)/(3 − (−1)) = −8/4 = −2.
Q10. The line y = (2/3)x + b passes through (6, 7). Find b.
A) 1 B) 2 C) 3 D) 4
Hint: 7 = (2/3)(6) + b = 4 + b → b = 3.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | B) 2 | Slope formula directly. |
Q2. | D) 7 | y-int = constant in slope-intercept form. |
Q3. | B) −3/2 | From Ax + By = C: m = −A/B. |
Q4. | C) y = (1/2)x − 5 | Same slope, different y-int. |
Q5. | D) y = −(1/5)x + 4 | Negative reciprocal of 5. |
Q6. | A) y = 4x − 3 | Plug into y = mx + b. |
Q7. | B) 2 | Substitute (3, 8) into equation. |
Q8. | D) y = −2 | Horizontal lines have form y = k. |
Q9. | A) −2 | Slope formula. |
Q10. | C) 3 | Plug in and solve for b. |