[1   2] [x] = [c]
  [1   2] [y]   [4]

Problem Setup

The system is:

This represents the equations x + 2y = c and x + 2y = 4.

Infinite Solutions Condition

For infinite solutions in a 2×2 system Ax = b, the coefficient matrix must have rank 1 (rows proportional), and the augmented matrix must match that rank.

Here, row 2 equals row 1, so rank is 1 if the right-hand sides satisfy c/1 = 4/1, or c = 4.

Verification

When c = 4, both equations become x + 2y = 4, a single line with solutions (x, y) = (4 - 2t, t) for any t.

If c ≠ 4, it’s inconsistent (parallel lines), yielding no solution.

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The value of c for infinite solutions in linear equations like x + 2y = c and x + 2y = 4 is a key concept in solving systems with dependent equations. This occurs when equations represent the same line.

Conditions Explained

Infinite solutions require proportional coefficients and constants: a₁/a₂ = b₁/b₂ = c₁/c₂.

Here, coefficients match (1=1, 2=2), so c/4 = 1, giving c = 4.

• Unique solution: Determinant nonzero.
• No solution: Coefficients proportional, constants not.
• Infinite solutions: All proportional (as here when c = 4).

Matrix Approach

Determinant 1·2 - 2·1 = 0, so singular. Augmented matrix rank equals coefficient rank only if c = 4.

Applications

Used in optimization and modeling where dependencies imply flexibility.