jmc

Question

"Modulo m graph paper" consists of a grid of m^2 points, representing all pairs of integer residues (x,y) where 0 x, y <m. To graph a congruence on modulo m graph paper, we mark every point (x,y) that satisfies the congruence. For example, a graph of y x^2 5 would consist of the points (0,0), (1,1), (2,4), (3,4), and (4,1). The graph of 3x 4y-1 35has a single x-intercept (x_0,0) and a single y-intercept (0,y_0), where 0 x_0,y_0<35. What is the value of x_0+y_0?

Accepted Answer

To find the x-intercept, we plug in 0 for y and solve 3x 4(0)-1 35.Multiplying both sides by 12, we get 36x -1235and thus x -1235. Translating this to the interval 0 x<35, we have x 2335, so the x-intercept on our graph is at (23,0). To find the y-intercept, we plug in 0 for x and solve 3(0) 4y-1 35.We can rewrite this as 1 4y35.Multiplying both sides by 9, we get 9 36y35,and thus y 935. So the y-intercept is at (0,9). We have x_0+y_0 = 23+9 = 32.