To determine the relationship between the two lines represented by the given system of linear equations, we'll follow these steps:
1. Convert each equation to the slope-intercept form (y = mx + b), where [tex]\( m \)[/tex] is the slope of the line.
2. Compare their slopes to find out if the lines are parallel, perpendicular, or neither.
Let's start with the first equation:
[tex]\[ \frac{1}{3} y = x - 9 \][/tex]
Multiply both sides by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 3(x - 9) \][/tex]
[tex]\[ y = 3x - 27 \][/tex]
The slope-intercept form of this equation is:
[tex]\[ y = 3x - 27 \][/tex]
So, the slope (m) of the first line is 3.
Now let’s examine the second equation:
[tex]\[ y = 3x - 3 \][/tex]
This is already in slope-intercept form, with:
[tex]\[ y = 3x - 3 \][/tex]
So, the slope (m) of the second line is also 3.
In conclusion:
- The slope of the first line is 3.
- The slope of the second line is 3.
Since the slopes are equal, the lines are parallel. Parallel lines are not perpendicular.
Therefore, the statement “The lines are perpendicular” is false. The correct interpretation is that the lines are parallel.
