Final answer:
To find whether the lines given by the equations 10 + 3x = 5y and 5x + 3y = 1 are parallel or perpendicular, we calculated the slopes (⅓ and -⅗ respectively) and found that they are neither equal nor negative reciprocals of each other; therefore, the lines are neither parallel nor perpendicular.
Step-by-step explanation:
To determine whether the pair of lines given by the equations 10 + 3x = 5y and 5x + 3y = 1 is parallel, perpendicular, or neither, we need to find the slopes of these lines to compare them.
Steps to Determine Line Relations
- Rewrite each equation in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
- For the first equation, 10 + 3x = 5y, divide everything by 5 to isolate y:
y = ⅓x + 2. The slope of the first line is ⅓.
- For the second equation, 5x + 3y = 1, rearrange the terms to solve for y:
3y = -5x + 1, and then divide all terms by 3:
y = -⅗x + ⅓. The slope of the second line is -⅗.
- Compare the slopes. If the slopes are equal, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular. In any other case, they are neither parallel nor perpendicular.
In this case, the slopes are ⅓ and -⅗, which are neither equal nor do they multiply to -1. Thus, the lines are neither parallel nor perpendicular to each other.