Lesson 3 • 45 min read
Linear Equations in Two Variables
In This Lesson
1Slope-Intercept Form
y = mx + b
m = slope (steepness)
How much y changes for each unit change in x
b = y-intercept
Where the line crosses the y-axis (when x = 0)
Example 1: y = 2x + 3
• Slope (m) = 2 (rises 2 for every 1 right)
• Y-intercept (b) = 3 (crosses y-axis at (0, 3))
Example 2: y = -½x + 4
• Slope (m) = -½ (falls ½ for every 1 right)
• Y-intercept (b) = 4 (crosses y-axis at (0, 4))
Converting to Slope-Intercept Form
2x + y = 5 ← Solve for y
y = -2x + 5 ← Subtract 2x from both sides
m = -2, b = 5
2Finding Slope
The Slope Formula
m = (y₂ - y₁) / (x₂ - x₁)
Rise over Run = Change in Y / Change in X
Remember:
RISE = vertical change (up/down)
RUN = horizontal change (left/right)
Tip:
Keep points in order! If you use y₂ first, you must use x₂ first in the denominator.
Example: Find slope through (2, 3) and (5, 9)
m = (y₂ - y₁) / (x₂ - x₁)
m = (9 - 3) / (5 - 2)
m = 6 / 3
m = 2
Example: Find slope through (-1, 4) and (3, -2)
m = (-2 - 4) / (3 - (-1))
m = -6 / 4
m = -3/2
3Types of Slopes
Positive Slope (m > 0)
↗️
Line goes UP from left to right
Example: y = 2x + 1
As x increases, y increases
Negative Slope (m < 0)
↘️
Line goes DOWN from left to right
Example: y = -3x + 5
As x increases, y decreases
Zero Slope (m = 0)
➡️
HORIZONTAL line
Example: y = 4
No change in y (flat line)
Undefined Slope
⬆️
VERTICAL line
Example: x = 3
Division by zero (run = 0)
Special Line Relationships
Parallel Lines
Same slope, different y-intercepts
y = 2x + 1 and y = 2x - 3
Both have m = 2 (never intersect)
Perpendicular Lines
Slopes are negative reciprocals
y = 2x + 1 and y = -½x + 4
m₁ × m₂ = -1 (intersect at 90°)
4Graphing Linear Equations
Method 1: Using Slope-Intercept Form
- Plot the y-intercept (b) on the y-axis
- From there, use the slope (m) to find another point
- Rise = numerator (move up if positive, down if negative)
- Run = denominator (always move right)
- Connect the points with a straight line
Example: Graph y = 2/3x - 2
1. Plot y-intercept: (0, -2)
2. From (0, -2), rise 2 and run 3 → (3, 0)
3. Draw line through (0, -2) and (3, 0)
Method 2: Using X and Y Intercepts
- Find y-intercept: Set x = 0, solve for y
- Find x-intercept: Set y = 0, solve for x
- Plot both intercepts and connect with a line
Example: Graph 2x + 3y = 6
Y-intercept: x = 0 → 3y = 6 → y = 2 → (0, 2)
X-intercept: y = 0 → 2x = 6 → x = 3 → (3, 0)
Draw line through (0, 2) and (3, 0)
5Point-Slope Form
y - y₁ = m(x - x₁)
Use when you know a point and the slope
Example 1: Write equation through (2, 5) with slope 3
y - y₁ = m(x - x₁)
y - 5 = 3(x - 2)
y - 5 = 3x - 6
y = 3x - 1
Example 2: Write equation through (-1, 4) and (3, -2)
Step 1: Find slope
m = (-2 - 4) / (3 - (-1)) = -6/4 = -3/2
Step 2: Use point-slope with either point
y - 4 = -3/2(x - (-1))
y - 4 = -3/2(x + 1)
y - 4 = -3/2x - 3/2
y = -3/2x + 5/2
When to Use Each Form
Slope-Intercept (y = mx + b): Best for graphing, given slope and y-intercept
Point-Slope (y - y₁ = m(x - x₁)): Best when given a point and slope, or two points
Standard Form (Ax + By = C): Best for finding intercepts, system of equations