What type of lines are 15x + 20y = 60 and 3x + 4y = 12?

Coincident

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.

This is a Class 10 Mathematics question. Chapter: Pair of Linear Equations in Two Variables. Topic: Graphical method of finding solutions..