To determine which line is parallel to the given equation [tex]\( y = -\frac{3}{5}x + \frac{1}{6} \)[/tex], we need to identify the slope of the line and then find another line from the given options that has the same slope.
The general form of a linear equation in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
1. For the given line [tex]\( y = -\frac{3}{5}x + \frac{1}{6} \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\( -\frac{3}{5} \)[/tex].
Now let's examine the slopes of the lines given in the options:
2. [tex]\( y = -\frac{3}{5}x - \frac{1}{4} \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\( -\frac{3}{5} \)[/tex].
3. [tex]\( y = \frac{3}{5}x - 4 \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\( \frac{3}{5} \)[/tex].
4. [tex]\( y = \frac{5}{3}x + 4 \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\( \frac{5}{3} \)[/tex].
5. [tex]\( y = -\frac{5}{3}x + \frac{1}{4} \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\( -\frac{5}{3} \)[/tex].
Two lines are parallel if they have identical slopes. Comparing the slopes, we can clearly see that the line:
[tex]\[ y = -\frac{3}{5}x - \frac{1}{4} \][/tex]
has the same slope ([tex]\( -\frac{3}{5} \)[/tex]) as the given line ([tex]\( y = -\frac{3}{5}x + \frac{1}{6} \)[/tex]).
Therefore, the line that is parallel to [tex]\( y = -\frac{3}{5}x + \frac{1}{6} \)[/tex] is:
[tex]\[ y = -\frac{3}{5}x - \frac{1}{4} \][/tex]
