ML Interview Q Series: How can we determine whether two linear equations in two variables have a unique solution, infinitely many solutions, or no solutions at all?

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Comprehensive Explanation

When dealing with two linear equations in two variables (commonly x and y), each equation can be expressed as:

a1 x + b1 y = c1 a2 x + b2 y = c2

where a1, b1, c1 and a2, b2, c2 are constants. One way to analyze the nature of solutions is by examining the determinant of the coefficient matrix. Specifically, define the determinant D as:

D captures how the two lines (represented by these equations) intersect in the coordinate plane.

• If D is not zero, the two lines intersect at exactly one point, giving a single unique solution. • If D equals zero, the two lines are either coincident (the same line) or parallel. – When the lines are coincident, every point on that line is a solution, thus infinitely many solutions exist. – When the lines are parallel, there is no intersection, so there is no solution.

In more detail, for the case D=0, we check the ratios a1/a2, b1/b2, and c1/c2 (assuming a2, b2, and c2 are not zero) to see whether they match. If the ratios a1/a2 and b1/b2 are the same as c1/c2, then the lines coincide. If not, the lines are parallel and have no intersection.

Further Insights

In practice, one often solves this system using linear algebra methods. For instance, in Python, we might do:

import numpy as np

# Example coefficients
a1, b1, c1 = 2, 3, 6
a2, b2, c2 = 1, -1, 1

A = np.array([[a1, b1],
              [a2, b2]])  # Coefficient matrix
B = np.array([c1, c2])     # Constant vector

# If the determinant of A is non-zero, we can solve for x, y
if np.linalg.det(A) != 0:
    solution = np.linalg.solve(A, B)
    print("Unique solution:", solution)
else:
    print("Either no solutions or infinitely many solutions.")

When np.linalg.det(A) != 0, there is a unique solution. If it is zero, it implies either infinite solutions or none, and additional checks are required to distinguish between these two cases.

Follow-Up Question: How do we distinguish between no solutions and infinitely many solutions in the case D=0?

If the determinant D=0, one should compare each ratio a1/a2, b1/b2, and c1/c2. If all these ratios are equal (and the denominators are not zero), then both equations describe the same line, implying infinitely many solutions. If these ratios are not the same, the lines are parallel and distinct, so there are no solutions.

Follow-Up Question: What if some coefficients are zero?

Even if certain coefficients (like a2=0 or b2=0) are zero, the same principle applies. You carefully check if the non-zero coefficients form consistent ratios. If all corresponding ratios match, expect infinitely many solutions. If they do not match, there will be none. If D is non-zero, a unique solution still exists.

Follow-Up Question: How do floating-point inaccuracies affect solution determination?

In numerical computation, floating-point precision might lead to a very small determinant, which is close to but not exactly zero. It becomes important to define a tolerance threshold. For instance, if |D| < 1e-12, one might treat D as effectively zero in practical applications. However, this might incorrectly classify a system as having infinite or no solutions if the threshold is not well-chosen. Robust methods or symbolic math can mitigate these numerical issues.

Follow-Up Question: Can this approach be generalized to more than two equations and two unknowns?

Yes. In higher dimensions, you form the coefficient matrix from the linear system. If the matrix is square and its determinant is non-zero, there is a unique solution. If the determinant is zero, you either get infinitely many solutions or no solution, depending on whether the equations remain consistent when you look at the augmented matrix (the original coefficient matrix with the constants appended as another column).