Answer:
The equation of a line parallel to y =2/3x - 4 that passes through the point (-6, 1) is [tex]\mathrm{y}=\frac{2}{3} \mathrm{x}+5[/tex] So option 1 is correct.
Solution:
Given, line equation is [tex]y=\frac{2}{3} x-4[/tex] and a point is (-6, 1).
We have to find the line equation of a line which is parallel to given line and passes through given point.
Now, let us find the slope of the given equation.
As we can see the given equation is in the form of slope intercept form which is given as
y = mx + c where m is slope.
So by comparison, slope = [tex]\frac{2}{3}[/tex]
We also know that, slopes of parallel lines are equal, then slope of our required line is [tex]\frac{2}{3}[/tex]
Then, let us find our line equation using point slope form,
[tex]\mathrm{y}-\mathrm{y}_{1}=\mathrm{m}\left(\mathrm{x}-\mathrm{x}_{1}\right)[/tex]
Here in our problem, [tex]\mathrm{m}=\frac{2}{3} \text { and }\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=(-6,1)[/tex]
Now, line equation ⇒ [tex]y-1=\frac{2}{3}(x-(-6))[/tex]
[tex]\begin{array}{l}{\rightarrow y-1=\frac{2}{3}(x+6)} \\\\ {\rightarrow 3(y-1)=2(x+6)} \\\\ {\rightarrow 3 y-3=2(x+6)} \\\\ {\rightarrow 3 y-3=2 x+12} \\\\ {\rightarrow 3 y=2 x+12+3} \\\\ {\rightarrow 3 y=2 x+15} \\\\ {\rightarrow y=\frac{2}{3} x+5}\end{array}[/tex]
Hence the equation of a line parallel to y =2/3x - 4 that passes through the point (-6, 1) is [tex]\mathrm{y}=\frac{2}{3} \mathrm{x}+5[/tex] So option 1 is correct.
