To determine whether the lines are parallel or perpendicular, we first need to find their slopes.
The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope. Let’s start with the first line:
1. First Line: y = 3x + 7
From this equation, we can see that the slope (m) is 3.
2. Second Line: 3x + 9y = 9
We need to rewrite this equation in slope-intercept form. Let’s start by isolating y.
9y = -3x + 9
y = - rac{1}{3}x + 1
Now, we can determine that the slope of the second line is -1/3.
Comparison of Slopes:
- First line slope (m1): 3
- Second line slope (m2): -1/3
For two lines to be parallel, their slopes must be equal. Since 3 is not equal to -1/3, the lines are not parallel.
For two lines to be perpendicular, the product of their slopes must equal -1. Let’s check:
3 * (-1/3) = -1
Since the product of the slopes is indeed -1, we can conclude that the lines are perpendicular.
In summary, the line y = 3x + 7 is perpendicular to the line 3x + 9y = 9.