Conditions for solvability

Multiply the first equation 7x − 4y = 29 by 3. This gives 21x − 12y = 87, which is precisely the second equation. Consequently, the equations are dependent and their graphs coincide. The ratio check gives 21/7 = (−12)/(−4) = 87/29 = 3, so the condition for infinitely many solutions is satisfied. Every point on the common line is a solution; therefore option C is correct, not the options for a unique or no solution.

For a pair of linear equations, let D be the determinant of the coefficients of x and y, while Dx and Dy are the auxiliary determinants. Cramer's rule gives a unique solution only when D is non-zero. If D = 0, the coefficient rows are dependent, so the represented lines are either coincident or parallel. When at least one auxiliary determinant is non-zero, the corresponding ratios are not all equal; hence the equations are inconsistent. Geometrically, the lines are distinct and parallel. They have no common point, so the pair has no solution. Therefore option C is correct, whereas infinitely many solutions require D = Dx = Dy = 0.

A pair of linear equations has no solution when the ratios of the coefficients of x and y are equal, but this common ratio is different from the ratio of the constant terms. Here, the x-coefficient ratio is (1/2)/2 = 1/4. To make the lines parallel, the y-coefficient ratio must also be q/8 = 1/4, which gives q = 2. The constant ratio is 3/5, not 1/4, so the equations are inconsistent and have no solution. Therefore option B is correct. Values q = 1, 4, or 8 do not make the two coefficient ratios equal, giving a unique intersection instead.

Answer: C, b = 4. A pair of linear equations has infinitely many solutions when one equation is an exact multiple of the other. Compare 2x + by = 6 with bx + 8y = 12. If b = 4, the first equation becomes 2x + 4y = 6. Multiplying it by 2 gives 4x + 8y = 12, which is exactly the second equation. Thus both equations represent the same straight line, so every point on that line is a solution and there are infinitely many solutions. Option A, b = 2, gives 2x + 2y = 6 and 2x + 8y = 12, which are different lines. Option B, b = 3, also does not make all corresponding coefficients proportional. Option D, b = 6, fails for the same reason. The ratio test is a useful check: a1/a2 = b1/b2 = c1/c2. Memory cue: infinitely many solutions means coincident lines, not merely intersecting lines.