Which of the following pairs of linear equations is/are consistent?A.…

The correct answer is Option (1) → A, B, C and D all are Consistent

A pair of linear equations is consistent when it has at least one solution. For \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\):

• \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\): the lines cross, one solution (consistent)

• \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\): the lines coincide, infinitely many solutions (consistent)

• \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\): the lines are parallel, no solution (inconsistent)

A. x + y − 5 = 0, 3x + 2y − 10 = 0 Consistent. \(\frac{1}{3}\ne\frac{1}{2}\), so the lines cross at one point.

B. 3x − 2y − 6 = 0, 6x − 4y − 12 = 0 Consistent. All three ratios are \(\frac{1}{2}\): the second equation is twice the first, the same line.

C. x − y − 9 = 0, 3x − 3y − 27 = 0 Consistent. All three ratios are \(\frac{1}{3}\): the second equation is three times the first.

D. x + y − 4 = 0, 3x + 3y − 12 = 0 Consistent. All three ratios are \(\frac{1}{3}\), so the two equations give the same line.

Tip: Coincident lines count as consistent; only parallel lines that never meet make a pair inconsistent.