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

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

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

✓ When you know two points on the line (calculate slope first)

✓ When you need to quickly write an equation without finding the y-intercept

Step 1: Start with the 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 the point-slope form

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

Example 1: Slope = 3, Point = (2, 5)

\[ y - 5 = 3(x - 2) \]

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

\[ y - (-1) = -2(x - 4) \quad \Rightarrow \quad y + 1 = -2(x - 4) \]

Example 3: Slope = \(\frac{1}{2}\), Point = (6, -3)

\[ y - (-3) = \frac{1}{2}(x - 6) \quad \Rightarrow \quad y + 3 = \frac{1}{2}(x - 6) \]

Example 4: Slope = \(-\frac{3}{4}\), Point = (-2, 7)

\[ y - 7 = -\frac{3}{4}(x - (-2)) \quad \Rightarrow \quad y - 7 = -\frac{3}{4}(x + 2) \]

Step 1: Calculate the 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 the slope and either point in the point-slope formula

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

Example: Find equation through (1, 3) and (4, 9)

Slope: \[ m = \frac{9-3}{4-1} = \frac{6}{3} = 2 \] Point-Slope Form: \[ y - 3 = 2(x - 1) \]

Example: Find equation through (-3, 2) and (2, -1)

Slope: \[ m = \frac{-1-2}{2-(-3)} = \frac{-3}{5} = -\frac{3}{5} \] Point-Slope Form: \[ y - 2 = -\frac{3}{5}(x - (-3)) \quad \Rightarrow \quad y - 2 = -\frac{3}{5}(x + 3) \]

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

Steps:
1. Distribute the slope on the right side
2. Add/subtract to isolate \(y\) on the left side

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

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

Example 2: Convert \(y + 3 = \frac{1}{2}(x - 6)\)

\[ y + 3 = \frac{1}{2}x - 3 \] \[ y = \frac{1}{2}x - 3 - 3 \] \[ y = \frac{1}{2}x - 6 \]

Goal: Convert to \(Ax + By = C\) form (where A, B, C are integers)

Steps:
1. Distribute and simplify
2. Move all terms with variables to the left side
3. Ensure coefficients are integers (multiply by LCD if needed)

Example: Convert \(y - 5 = 2(x - 3)\) to standard form

\[ y - 5 = 2x - 6 \] \[ y = 2x - 1 \] \[ -2x + y = -1 \] \[ 2x - y = 1 \quad \text{(multiply by -1)} \]

Horizontal Line (slope = 0)

Point-Slope: \[ y - y_1 = 0(x - x_1) \] Simplifies to: \[ y = y_1 \] Example: Line through (3, 5) with slope 0: \(y = 5\)

Vertical Line (undefined slope)

Cannot use point-slope form!
Equation: \[ x = x_1 \] Example: Line through (4, 7) that's vertical: \(x = 4\)

Line Through Origin \((0, 0)\)

Point-Slope: \[ y - 0 = m(x - 0) \] Simplifies to: \[ y = mx \] Example: Slope 3 through origin: \(y = 3x\)

Point-Slope Form:

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

Use when: You know slope and one point

Slope-Intercept Form:

\[ y = mx + b \]

Use when: You know slope and y-intercept

Standard Form:

\[ Ax + By = C \]

Use when: You need integer coefficients or x and y-intercepts

Problem Type 1: Given slope and one point

Step 1: Identify \(m\), \(x_1\), and \(y_1\)
Step 2: Substitute into \(y - y_1 = m(x - x_1)\)
Step 3: Simplify if needed

Problem Type 2: Given two points

Step 1: Calculate slope: \(m = \frac{y_2-y_1}{x_2-x_1}\)
Step 2: Choose either point as \((x_1, y_1)\)
Step 3: Substitute into \(y - y_1 = m(x - x_1)\)
Step 4: Simplify if needed

❌ Mistake: Forgetting to change sign when subtracting negative numbers
✅ Correct: \(y - (-3) = y + 3\), not \(y - 3\)

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

❌ Mistake: Subtracting in wrong order when finding slope
✅ Correct: \(m = \frac{y_2-y_1}{x_2-x_1}\), not \(\frac{y_1-y_2}{x_2-x_1}\)

❌ Mistake: Using point-slope form for vertical lines
✅ Correct: Use \(x = x_1\) instead

Main Formula:

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

Slope Formula (for two points):

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

Remember:
• \(m\) = slope (rise over run)
• \((x_1, y_1)\) = known point on the line
• Works for any non-vertical line
• Can easily convert to other forms