Question

Vocabulary How can you determine the number of solutions
of a system by looking at the equations?

Answer 1

Short answer: look at the coefficients. Use slopes/determinant for 2×2 systems or rank for larger systems.

Two linear equations in two variables (ax + by = c and a' x + b' y = c'):
- Compute the determinant D = a b' − a' b (or compare slopes a/b and a'/b').
- If D ≠ 0 (slopes different) → exactly one solution (the lines intersect at one point).
- If D = 0 and a/a' = b/b' = c/c' (all three ratios equal) → infinitely many solutions (the equations are the same line).
- If D = 0 but a/a' = b/b' ≠ c/c' (left sides proportional, right sides not) → no solution (parallel distinct lines).

Three or more linear equations / more variables:
- Put the system into matrices and use Gaussian elimination to find the ranks.
- Let A be the coefficient matrix and [A|b] the augmented matrix.
- If rank(A) = rank([A|b]) = number of unknowns → unique solution.
- If rank(A) = rank([A|b]) < number of unknowns → infinitely many solutions (free parameters).
- If rank(A) < rank([A|b]) → no solution (inconsistent).

Vocabulary:
- consistent = has at least one solution; inconsistent = no solution.
- independent = equations are not multiples of each other (can lead to a unique solution); dependent = some equations are multiples/linear combinations of others (can lead to infinitely many solutions).

If you want, give me a specific system and I’ll tell you which case it is.