GATE CS 1987 Q1-xxi — Linear Algebra: Matrices, Determinants, System of Linear Equations

A linear inequality in two variables $x$ and $y$ is an inequality that can be written in one of the following forms:

where $A, B,$ and $C$ are constants, and the variables $x$ and $y$ are of first degree (exponent of $1$) with no product terms like $xy$.

Let's evaluate each option given that $a, b,$ and $c$ are constants:

• Option A: $ax + bcy = 0$
  This is a linear equation, not an inequality.
• Option B: $ax^2 + cy^2 = 21$
  This is a non-linear equation due to the $x^2$ and $y^2$ terms.
• Option C: $abx + a^2y \ge 15$
  Since $a$ and $b$ are constants, the coefficients $ab$ and $a^2$ are also constants. This inequality is of the form $Ax + By \ge C$, which is a linear inequality.
• Option D: $xy + ax \ge 20$
  This is non-linear because of the product term $xy$ (which has a degree of 2).

Therefore, the correct option is C.