Final answer:
To determine if two linear equations are parallel, we can compare their slopes. The equations 3x - 4y = -1 and 3y + 9x = -6 have different slopes when rewritten in the slope-intercept form, 3/4 and 3 respectively, and thus are not parallel.
Step-by-step explanation:
To determine whether the graphs of 3x - 4y = -1 and 3y + 9x = -6 are parallel without doing any calculations, we can look at the equations to see if they can be written in slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept. If both equations can be manipulated to have the same slope 'm', then the lines are parallel.
Let's rewrite the equations in the slope-intercept form using mental math:
- For the first equation, 3x - 4y = -1, the 'y' term needs to be isolated. Mentally, we would divide every term by -4 to get y by itself, leading to a slope (m) of 3/4.
- For the second equation, 3y + 9x = -6, dividing every term by 3 to isolate 'y' would lead to a slope (m) of 9/3 or 3.
However, since the slopes are different (3/4 for the first equation and 3 for the second), the lines are not parallel. Thus, the correct answer is (b) They have different slopes.