linear equations

Examples of linear equations:
- 3x + 7 = 22
- -2x = 14
- x/4 - 3 = 5
- 0.5x + 1.2 = 3.7

Non-linear equations you'll encounter later:
- x² + 3 = 12 (quadratic)
- 2/x = 8 (rational)
- √x = 4 (radical)

Addition ↔ Subtraction: If something's added to x, subtract it from both sides. If something's subtracted from x, add it to both sides.

Multiplication ↔ Division: If x is multiplied by a number, divide both sides by that number. If x is divided by a number, multiply both sides by that number.

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

When you have two equations like $2x + 3y = 7$ and $x - y = 1$, you're looking for the values of $x$ and $y$ that make BOTH equations true at the same time. Think of it as finding coordinates $(x, y)$ that satisfy both conditions simultaneously.

Every system has exactly one of three outcomes:
- One solution: The lines intersect at exactly one point
- No solution: The lines are parallel (same slope, different y-intercepts)
- Infinitely many solutions: The lines are identical (same slope, same y-intercept)

You solve linear inequalities almost exactly like linear equations. Add, subtract, multiply, and divide to isolate your variable. There's just one crucial rule: when you multiply or divide both sides by a negative number, flip the inequality sign.

Let's see why this matters:
- Start with: 5 > 3 (true)
- Multiply both sides by -1: -5 > -3 (false!)
- Flip the sign: -5 < -3 (true again)

When you have multiple linear equations working together, you're dealing with a system. The key insight? You're looking for values that satisfy ALL equations simultaneously.

Consider this system:

You have three methods to solve this:

Substitution: Solve one equation for a variable, then substitute into the other. From the second equation: x = y + 1. Substitute into the first: 2(y + 1) + 3y = 12. This gives you 2y + 2 + 3y = 12, so 5y = 10, and y = 2. Then x = 2 + 1 = 3.