\[ y - y_1 = m(x - x_1) \]

Where:
• \(m\) = slope of the line
• \((x_1, y_1)\) = a known point on the line
• \((x, y)\) = any other point on the line (variables)

✓ When you know the slope and one point on the line

✓ When you know two points (calculate slope first)

✓ When you need an equation without finding y-intercept

Step 1: Start with slope formula

\[ m = \frac{y - y_1}{x - x_1} \]

Step 2: Multiply both sides by \((x - x_1)\)

\[ m(x - x_1) = y - y_1 \]

Step 3: Rewrite to get point-slope form

\[ y - y_1 = m(x - x_1) \]

Example 1: Slope = 3, Point = (2, 4)

\[ y - 4 = 3(x - 2) \]

Example 2: Slope = -5, Point = (-4, 7)

\[ y - 7 = -5(x - (-4)) \quad \Rightarrow \quad y - 7 = -5(x + 4) \]

Example 3: Slope = \(\frac{1}{2}\), Point = (8, 2)

\[ y - 2 = \frac{1}{2}(x - 8) \]

Example 4: Slope = \(\frac{3}{4}\), Point = (4, -6)

\[ y - (-6) = \frac{3}{4}(x - 4) \quad \Rightarrow \quad y + 6 = \frac{3}{4}(x - 4) \]

Step 1: Calculate slope using two points \((x_1, y_1)\) and \((x_2, y_2)\)

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 2: Use slope and either point in formula

\[ y - y_1 = m(x - x_1) \]

Example: Points (1, 10) and (3, 16)

Slope: \[ m = \frac{16-10}{3-1} = \frac{6}{2} = 3 \] Point-Slope: \[ y - 10 = 3(x - 1) \]

Example: Points (8, -3) and (-2, 6)

Slope: \[ m = \frac{6-(-3)}{-2-8} = \frac{9}{-10} = -\frac{9}{10} \] Point-Slope: \[ y - (-3) = -\frac{9}{10}(x - 8) \quad \Rightarrow \quad y + 3 = -\frac{9}{10}(x - 8) \]

Goal: Convert to \(y = mx + b\) form

Steps:
1. Distribute slope on right side
2. Isolate \(y\) on left side

Example 1: Convert \(y - 4 = 3(x - 2)\)

\[ y - 4 = 3x - 6 \] \[ y = 3x - 6 + 4 \] \[ y = 3x - 2 \]

Example 2: Convert \(y + 5 = \frac{2}{3}(x - 3)\)

\[ y + 5 = \frac{2}{3}x - 2 \] \[ y = \frac{2}{3}x - 2 - 5 \] \[ y = \frac{2}{3}x - 7 \]

Goal: Convert to \(Ax + By = C\) form

Steps:
1. Distribute and simplify
2. Move variables to left side
3. Ensure integer coefficients

Example: Convert \(y - 7 = -5(x + 4)\)

\[ y - 7 = -5x - 20 \] \[ y = -5x - 13 \] \[ 5x + y = -13 \]

Horizontal Line (slope = 0)

\[ y - y_1 = 0(x - x_1) \] Simplifies to: \[ y = y_1 \]

Vertical Line (undefined slope)

Cannot use point-slope form!
Use: \[ x = x_1 \]

Line Through Origin

\[ y - 0 = m(x - 0) \] Simplifies to: \[ y = mx \]

Point-Slope Form:

\[ y - y_1 = m(x - x_1) \]

Use when: slope and one point known

Slope-Intercept Form:

\[ y = mx + b \]

Use when: slope and y-intercept known

Standard Form:

\[ Ax + By = C \]

Use when: integer coefficients needed

❌ Forgetting sign change: \(y - (-3)\)
✅ Correct: \(y + 3\)

❌ Wrong slope calculation order
✅ Correct: \(m = \frac{y_2-y_1}{x_2-x_1}\)

❌ Mixing up coordinates
✅ Point (3, 5): \(x_1 = 3\), \(y_1 = 5\)

\[ y - y_1 = m(x - x_1) \]

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]