Which of the following pairs of equations represents parallel lines?
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2x + 2y = 8, 2x + y = 7
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x – y = 17, 2x – 2y = 38
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2x + y = 3, 5x + 2y = 3
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x + y = 12, x – y = 0
For two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 to be parallel, $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$. In lines x – y = 17 and 2x – 2y = 38, $\frac{a_1}{a_2} = \frac{1}{2}, \frac{b_1}{b_2} =\frac{-1}{-2} = \frac{1}{2} , \frac{c_1}{c_2} = \frac{3}{12} = \frac{17}{38}$ Therefore, $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ Hence, the lines are parallel.
Parallel lines have the same slope but different intercepts. Rewriting x - y = 17 as x - y = 17 and 2x - 2y = 38 as x - y = 19 (dividing by 2) shows both have slope 1 but different constants, so they never meet - true parallel lines. The other pairs either reduce to the same line (no distinct parallel pair) or have different slopes entirely, so they intersect instead.