Which of the following pairs of equations represents parallel lines?
-
x – y = 7, 2x – 2y = 15
-
x + 2y = 1, y + 3x = 4
-
2x + y = 4, x – 2y = 3
-
x + 2y = 4, 3x + 7y = 18
Two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are parallel when $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$. In lines x – y = 7 and 2x – 2y = 15, $\frac{a_1}{a_2} = \frac{1}{2}, \frac{b_1}{b_2} =\frac{-1}{-2} , \frac{c_1}{c_2} = \frac{7}{15}$ $\therefore$ $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
Parallel (non-coincident) lines have equal ratios of x and y coefficients but a different ratio for the constant term: a1/a2 = b1/b2 ≠ c1/c2. For x–y=7 and 2x–2y=15, the coefficient ratio is 1/2 = -1/-2, but 7/15 is not equal to 1/2, so the lines are parallel and never intersect.