Graphical method of finding solutions.

The governing concept is testing whether a point lies on a line by substitution. A point (x, y) lies on 5x + 11y = 55 only when its coordinates make the left-hand side equal to 55. For (11, 0), 5(11) + 11(0) = 55, so it lies on the line. For (0, 5), 5(0) + 11(5) = 55, so it also lies on the line. For (22/5, 3), 5(22/5) + 11(3) = 22 + 33 = 55, so this point is valid as well. For (4, 3), the value is 5(4) + 11(3) = 20 + 33 = 53, not 55. Therefore option D is the only point that does not lie on the line.

The governing concept is identifying a pair of lines by comparing the ratios of their coefficients. Rewrite the first equation by dividing every term by 5: 15x + 20y = 60 becomes 3x + 4y = 12. This is exactly the second equation, not merely a proportional equation with a different constant. Equivalently, 15/3 = 20/4 = 60/12 = 5. Equal ratios for all three corresponding coefficients mean that both equations describe the same geometric line. Such lines are called coincident lines, and every point on one is also on the other, giving infinitely many common solutions. Therefore option C is correct. Intersecting lines have unequal slope ratios, distinct parallel lines have proportional x and y coefficients but a different constant ratio, and perpendicular lines require slopes whose product is −1.

The coefficient ratios \\(2/5\\) and \\(9/11\\) are unequal. Therefore, the two lines have different slopes and intersect at exactly one point, giving a unique solution. Option B would be correct only if the lines were distinct and parallel. Exam tip: for \\(a_1x+b_1y+c_1=0\\) and \\(a_2x+b_2y+c_2=0\\), if \\(a_1/a_2 \\ne b_1/b_2\\), the pair has a unique solution.