What Is a System of Linear Equations?
A system of linear equations consists of two or more linear equations sharing the same set of variables. Solving the system means finding values for the variables that satisfy all equations simultaneously.
Method 1: Substitution
The substitution method works best when one equation can easily be solved for a single variable.
Step-by-Step Example
Solve the system:
- 2x + y = 7
- x − y = 2
Step 1: Solve equation 2 for x → x = y + 2.
Step 2: Substitute x into equation 1: 2(y + 2) + y = 7 → 2y + 4 + y = 7 → 3y = 3 → y = 1.
Step 3: Plug y back in: x = 1 + 2 = 3. Solution: (x = 3, y = 1).
Method 2: Elimination (Linear Combination)
Elimination involves adding or subtracting equations to eliminate one of the variables.
Method 3: Matrix Reduced Row Echelon Form (RREF)
For systems with 3 or more variables (e.g. 3x3 or 4x4), manual substitution becomes tedious. Representing the system as an augmented matrix [A|b] and applying Gaussian Elimination to transform it into Reduced Row Echelon Form (RREF) guarantees an exact solution.