\[ y = mx + b \]

Where:
• \(y\) = y-coordinate of any point on the line
• \(m\) = slope of the line (rise over run)
• \(x\) = x-coordinate of any point on the line
• \(b\) = y-intercept (where line crosses y-axis)

Note: Some textbooks use \(y = mx + c\) where \(c\) is the y-intercept. Both forms are equivalent!

Slope Formula:

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

Types of Slopes:
• Positive slope (\(m > 0\)): Line rises from left to right
• Negative slope (\(m < 0\)): Line falls from left to right
• Zero slope (\(m = 0\)): Horizontal line
• Undefined slope: Vertical line (cannot use slope-intercept form)

• The y-intercept is the point where the line crosses the y-axis
• At this point, \(x = 0\)
• Written as the ordered pair \((0, b)\)
• The value \(b\) can be positive, negative, or zero

Examples:

• If \(b = 3\), the line crosses y-axis at \((0, 3)\)
• If \(b = -2\), the line crosses y-axis at \((0, -2)\)
• If \(b = 0\), the line passes through the origin \((0, 0)\)

Start with: Standard form \(Ax + By + C = 0\)

\[ Ax + By + C = 0 \]

Step 1: Isolate \(By\)

\[ By = -Ax - C \]

Step 2: Divide by \(B\)

\[ y = -\frac{A}{B}x - \frac{C}{B} \]

Result: Slope-intercept form where \(m = -\frac{A}{B}\) and \(b = -\frac{C}{B}\)

\[ y = mx + b \]

Example 1: Slope = 2, y-intercept = 5

\[ y = 2x + 5 \]

Example 2: Slope = -3, y-intercept = 7

\[ y = -3x + 7 \]

Example 3: Slope = \(\frac{1}{2}\), y-intercept = -4

\[ y = \frac{1}{2}x - 4 \]

Example 4: Slope = \(-\frac{3}{4}\), y-intercept = \(\frac{1}{5}\)

\[ y = -\frac{3}{4}x + \frac{1}{5} \]

Step 1: Identify the y-intercept (where line crosses y-axis)

This gives you the value of \(b\)

Step 2: Find the slope

• Choose two points on the line
• Count the rise (vertical change)
• Count the run (horizontal change)
• Calculate: \(m = \frac{\text{rise}}{\text{run}}\)

Step 3: Write the equation

Substitute values into \(y = mx + b\)

Example: Line crosses y-axis at (0, -2), rises 3 units for every 1 unit right

\(b = -2\), \(m = \frac{3}{1} = 3\)
Equation: \(y = 3x - 2\)

Step 1: Calculate the slope

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

Step 2: Find b by substituting one point and slope into \(y = mx + b\)

\[ b = y - mx \]

Example: Find equation through (1, 5) and (-4, 7)

Step 1: \[ m = \frac{7-5}{-4-1} = \frac{2}{-5} = -\frac{2}{5} \] Step 2: Using (1, 5): \[ 5 = -\frac{2}{5}(1) + b \Rightarrow b = 5 + \frac{2}{5} = \frac{27}{5} \] Equation: \[ y = -\frac{2}{5}x + \frac{27}{5} \]

Example 1: Convert \(3x + 4y + 5 = 0\) to slope-intercept form

\[ 4y = -3x - 5 \] \[ y = -\frac{3}{4}x - \frac{5}{4} \] Slope: \(m = -\frac{3}{4}\), Y-intercept: \(b = -\frac{5}{4}\)

Example 2: Convert \(2x - y = 6\) to slope-intercept form

\[ -y = -2x + 6 \] \[ y = 2x - 6 \] Slope: \(m = 2\), Y-intercept: \(b = -6\)

Example: Convert \(y = 3x - 2\) to standard form \(Ax + By = C\)

\[ y = 3x - 2 \] \[ -3x + y = -2 \] \[ 3x - y = 2 \quad \text{(multiply by -1)} \]

Horizontal Line (slope = 0)

\[ y = 0x + b \quad \Rightarrow \quad y = b \] Example: \(y = 5\) (horizontal line through y = 5)

Line Through Origin (y-intercept = 0)

\[ y = mx + 0 \quad \Rightarrow \quad y = mx \] Example: \(y = 4x\) (line through origin with slope 4)

Vertical Line (undefined slope)

Cannot use slope-intercept form!
Use: \(x = c\) where \(c\) is the x-coordinate

Parallel Lines

• Have the same slope
• Different y-intercepts
Example: \(y = 3x + 5\) is parallel to \(y = 3x - 1\)

Perpendicular Lines

• Slopes are negative reciprocals
• If one slope is \(m\), the other is \(-\frac{1}{m}\)
Example: \(y = 2x + 3\) is perpendicular to \(y = -\frac{1}{2}x + 1\)

Step 1: Plot the y-intercept

Plot point \((0, b)\) on the y-axis

Step 2: Use the slope to find another point

• From the y-intercept, move according to rise/run
• If \(m = \frac{2}{3}\): rise 2, run 3
• If \(m = -4 = \frac{-4}{1}\): rise -4 (down 4), run 1

Step 3: Draw the line

Connect the two points with a straight line

❌ Confusing slope and y-intercept
✅ Remember: \(m\) = slope, \(b\) = y-intercept

❌ Writing \(y = b + mx\) instead of \(y = mx + b\)
✅ Always write slope term first: \(y = mx + b\)

❌ Forgetting negative signs
✅ \(y = 2x - 5\) not \(y = 2x + -5\)

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

Slope-Intercept Form:

\[ y = mx + b \]

Best for: Graphing, identifying slope and y-intercept quickly

Point-Slope Form:

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

Best for: Writing equation when you know slope and one point

Standard Form:

\[ Ax + By = C \]

Best for: Finding x and y intercepts, integer coefficients

\[ y = mx + b \]

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

Key Points:
• \(m\) = slope (steepness of line)
• \(b\) = y-intercept (where line crosses y-axis)
• Most common form for graphing
• Cannot be used for vertical lines