Determine which of the lines, if any, are parallel. explain. 20 points line a: 3y−x=6 line b: 3y=x+18 line c: 3y−2x=9

a and b are parallel

When comparing lines to see which are parallel or perpendicular, it is conventient to have their equations in the same form.

Lines a and c are not parallel, because the coefficients of x and y are different.

To compare these against line b, we can subtract x to put the equation in the same form as the others:

... 3y -x = 18

This lets us see that the coefficients of x and y match those of line a, so the lines (a and b) are parallel.

Comment on the equations

Here, we're comparing the equations in "standard form" with the variable in the leading term being y. An equation for a line written in standard form has integer coefficients that are relatively prime. That is, the equation of the line (line d) ...

... 3y -x = 6.1

would be written in standard form as

... 30y -10x = 61

This line is parallel to the lines of a and b, but has different coefficients for x and y. So, what is important is that the ratio of coefficients be the same, not their specific values. Effectively, if we solve for y, we're looking at the coefficient of x (the slope) to see if the lines are parallel.

... line a: y = (1/3)x + 2

... line b: y = (1/3)x + 6

... line c: y = (2/3)x + 3 . . . . . x-coefficient (slope) is different from a, b

... line d: y = (1/3)x + 61/30 . . . . . lines a, b, d are parallel

44

step-by-step explanation: