Systems of linear equations · Grade 12

In Grade 12 you learn: systems of linear equations.

A linear system is a set of equations, like 2x + y − z = 3, that must all be true at once. With two unknowns, each equation is a line. With three, each one is a plane. Elimination, back-substitution and matrices solve them. The determinant tells you in advance whether there is one solution, none, or infinitely many.

Key facts

• Write the system as AX = B. If det A ≠ 0, there is exactly one solution: X = A^(−1)B.
• Gaussian elimination: use row operations such as R₂ → R₂ − kR₁ to reach triangular form. Then back-substitute.
• If det A = 0 there are either no solutions (inconsistent) or infinitely many (consistent).
• Two equations in two unknowns: crossing lines (one solution), parallel lines (none), or the same line (infinitely many).
• Three planes: one common point, a common line (a sheaf), or no common point (a triangular prism, or parallel planes).
• Always check a solution in every original equation.