Linear Equations in Two Variables: Graphing and Properties

TL;DR

Linear equations in two variables create straight lines when graphed. You'll learn to identify slope, y-intercept, and different forms of these equations. Every straight line represents infinitely many solutions to the equation.

1. The Mental Model

A linear equation with two variables is like a recipe that connects x and y values. For every x you pick, the equation tells you exactly what y must be. Plot enough of these (x,y) pairs and you'll always get a perfectly straight line. That's the whole idea.

2. The Core Material

Standard Form and Slope-Intercept Form

Linear equations come in different forms, but they're all describing the same straight line. The most common form you'll work with is slope-intercept form: y = mx + b.

In this form, m is the slope (how steep the line is) and b is the y-intercept (where the line crosses the y-axis). For example, y = 3x + 2 has a slope of 3 and crosses the y-axis at (0, 2).

Standard form looks like Ax + By = C, where A, B, and C are integers. The equation 2x + 3y = 12 is in standard form. You can convert between forms using algebra. To go from standard to slope-intercept, solve for y:
2x + 3y = 12
3y = -2x + 12
y = -⅔x + 4

Now you can see the slope is -⅔ and the y-intercept is 4.

Understanding Slope

Slope measures how much y changes when x increases by 1. It's the "rise over run." A slope of 3 means for every step right, you go up 3 steps. A slope of -½ means for every 2 steps right, you go down 1 step.

You can calculate slope between any two points (x₁, y₁) and (x₂, y₂) using: m = (y₂ - y₁)/(x₂ - x₁)

Positive slopes go upward from left to right. Negative slopes go downward. A slope of zero is horizontal. An undefined slope (division by zero) is vertical.

Graphing Strategies

To graph a linear equation, you need at least two points. Here are the fastest methods:

Method 1: Use slope and y-intercept. Start at the y-intercept, then use the slope to find your next point. If y = 2x - 1, start at (0, -1), then go right 1 and up 2 to reach (1, 1).

Method 2: Find intercepts. Set x = 0 to find the y-intercept, then set y = 0 to find the x-intercept. For 2x + 3y = 12: when x = 0, y = 4. When y = 0, x = 6. Plot (0, 4) and (6, 0), then connect them.

Method 3: Make a table. Pick any x-values, calculate the corresponding y-values, and plot those points.

graph LR
    A["Pick two points"] --> B["Plot on coordinate plane"]
    B --> C["Draw straight line through them"]
    C --> D["Extend line in both directions"]

Special Cases

Horizontal lines have equations like y = 5 (slope is zero). Vertical lines have equations like x = -2 (slope is undefined). These are still linear equations, just special ones.

Parallel lines have the same slope but different y-intercepts. Perpendicular lines have slopes that multiply to -1 (negative reciprocals).

3. Worked Example

Let's graph the equation 3x - 2y = 6 and identify its properties.

First, I'll convert to slope-intercept form:
3x - 2y = 6
-2y = -3x + 6
y = (3/2)x - 3

Now I can see the slope is 3/2 and the y-intercept is -3.

To graph this, I'll start at the y-intercept (0, -3). Since the slope is 3/2, I'll move right 2 units and up 3 units to get my next point: (2, 0).

Let me verify: when x = 2, y = (3/2)(2) - 3 = 3 - 3 = 0. ✓

I could also find the x-intercept by setting y = 0:
0 = (3/2)x - 3
3 = (3/2)x
x = 2

So the x-intercept is (2, 0), which matches what I found using the slope.

Drawing a line through (0, -3) and (2, 0) gives me the complete graph. The line rises from left to right because the slope is positive.

4. Key Takeaways

4.1 Most Important Concepts

• Every linear equation graphs as a straight line — no curves, no bends, perfectly straight
• Slope tells you the line's steepness and direction — positive goes up, negative goes down
• Y-intercept is where the line crosses the y-axis — always has x-coordinate of zero
• Two points determine a unique line — you only need two points to graph any linear equation
• Parallel lines have identical slopes — they never intersect because they rise at the same rate
• Standard form Ax + By = C works for all lines — even vertical ones that slope-intercept can't handle
• Solutions are all the (x,y) pairs on the line — infinitely many points satisfy the equation

4.2 Common Misconceptions

• "Slope is always rise/run" — Actually, it's change in y over change in x, which could be negative
• "Linear equations always have two variables" — Equations like y = 5 are linear with effectively one variable
• "Steeper lines have smaller slopes" — Steepness depends on absolute value; a slope of -10 is steeper than 2
• "The line stops at the points you plot" — Lines extend infinitely in both directions

4.3 Compare & Contrast

5. Now Try It

Graph the equation 4x + 2y = 8 using two different methods. First, convert to slope-intercept form and use the slope and y-intercept. Second, find both x and y intercepts and connect them. Verify that both methods give you the same line.

Success looks like: You get the same straight line both ways, passing through (0, 4) and (2, 0), with a slope of -2.