Answer with explanation:
Equation of a line passing through point (a,b) and has a slope m is given by :-
[tex](y-b)=m(x-a)[/tex]
From the given picture , given point = (-3,2)
a) Equation of a line passing through point (-3 ,2) and has a slope 3 is given by :-
[tex](y-(2))=3(x-(-3))\\\\\Rightarrow\ y-2=3(x+3)\\\\\Rightarrow\ y=3x+9+2\\\\\Rightarrow\ y=3x+11[/tex]
b) Equation of a line passing through point (-3 ,2) and has a slope -5 is given by :-
[tex](y-(2))=-5(x-(-3))\\\\\Rightarrow\ y-2=-5(x+3)\\\\\Rightarrow\ y=-5x-15+2\\\\\Rightarrow\ y=-5x-13[/tex]
c) Equation of a line passing through point (-3 ,2) and has a slope [tex]\dfrac{2}{3}[/tex]is given by :-
[tex](y-(2))=\dfrac{2}{3}(x-(-3))\\\\\Rightarrow\ y-2=\dfrac{2}{3}(x+3)\\\\\Rightarrow\ y=\dfrac{2}{3}x+2+2\\\\\Rightarrow\ y=\dfrac{2}{3}x+4[/tex]
We know that when slope is undefined , then the line is parallel to y axis.
d) The line parallel to y axis and passes through given point : x=-3
[Note , when slope is undefined , it means x-values does not change through out the line ∵ slope = [tex]\dfrac{\text{Change in y values}}{\text{Change in x-values}}[/tex]]
