A single equation is like a clue: "The sum of two numbers is 10." There are infinite answers (5+5, 1+9, -2+12).
But if you add a second clue: "The difference is 2," suddenly there is only one answer (6 and 4). This is a System of Equations.

1. The Visual Approach (Intersection)

Every linear equation (y = mx + b) is a straight line on a graph.
The solution to the system is simply the point where the two lines cross.
This is why systems are so important: they represent the "sweet spot" that satisfies everyone's constraints.

2. Method 1: The Substitution Method

Best used when one variable is already alone (e.g., y = 2x + 1).

• Step 1: Isolate one variable.
• Step 2: PLUG IT IN to the other equation. Now you have an equation with only one variable. Solve it.
• Step 3: Plug that answer back in to find the first variable.

Example:
1. x + y = 10
2. y = x - 2
Substitute (x-2) for y in eqn 1:
x + (x-2) = 10 -> 2x = 12 -> x = 6.

3. Method 2: The Elimination Method

Best used when equations are "stacked" (standard form). You add or subtract the equations to KILL a variable.

Example:
2x + y = 10
-2x + y = 4
------------
0x + 2y = 14 -> y = 7.

We eliminated x instantly! This is the method computer algorithms use (Gaussian Elimination).

4. Method 3: Matrices (The Nuclear Option)

If you have 2 equations, substitution is fine. If you have 100 equations with 100 variables, you need Linear Algebra.

We write the system as Ax = B.
• A = Matrix of coefficients.
• x = Vector of variables.
• B = Vector of answers.

To solve, we multiply by the inverse: x = A⁻¹ * B. This is how GPS satellites calculate your position using signals from 4 different satellites.

5. Weird Cases: Parallel Lines

Sometimes, math breaks.

• No Solution (Parallel Lines): x + y = 5 and x + y = 10. Two numbers cannot sum to 5 and 10 at the same time. The lines never cross.
• Infinite Solutions (Same Line): x + y = 5 and 2x + 2y = 10. They look different, but line 2 is just line 1 doubled. They are the same line. Every point on the line is a solution.

6. Real World Applications

• Business (Break-Even): Cost = 50 + 2x. Revenue = 4x. Set them equal to find how many items you must sell to start making money.
• Chemistry (Balancing Equations): You have H2 and O2 making H2O. You set up a system of linear equations to make sure the number of atoms matches on both sides.

7. FAQ

Q: Can I have 3 variables but only 2 equations?
A: No. You need at least as many equations as you have variables to find a unique solution. (N equations for N unknowns). If you have fewer, you get a "Line of solutions" (infinite).

Q: What is a Homogeneous System?
A: A system where all equations equal zero. It always has at least the solution (0,0,0), which we call the "Trivial Solution".

Conclusion

Solving systems is the art of constraint satisfaction. It is finding the single path that obeys every rule at once.