Answer-
The line equation is,
[tex]\boxed{\boxed{6x+15y-10=0}}[/tex]
Solution-
The line [tex]y =3x-5[/tex] meets x-axis at the point M, i.e M is the x-intercept of this line. At the x-intercept y=0, so
[tex]\Rightarrow 0 =3x-5[/tex]
[tex]\Rightarrow 3x=5[/tex]
[tex]\Rightarrow x=\dfrac{5}{3}[/tex]
So, coordinate of M is [tex](\dfrac{5}{3},\ 0)[/tex]
The line [tex]3y+2x=2[/tex] meets y-axis at point N, i.e N is the y-intercept of this line. At the y-intercept x=0, so
[tex]\Rightarrow 3y+2(0)=2[/tex]
[tex]\Rightarrow 3y=2[/tex]
[tex]\Rightarrow y=\dfrac{2}{3}[/tex]
So, coordinate of N is [tex](0,\ \dfrac{2}{3})[/tex]
The line joining M and N can be found out by applying two point formula of straight line,
[tex]\Rightarrow \dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}[/tex]
[tex]\Rightarrow \dfrac{y-0}{\frac{2}{3}-0}=\dfrac{x-\frac{5}{3}}{0-\frac{5}{3}}[/tex]
[tex]\Rightarrow \dfrac{y}{\frac{2}{3}}=\dfrac{x-\frac{5}{3}}{-\frac{5}{3}}[/tex]
[tex]\Rightarrow -\dfrac{5}{3}y=\dfrac{2}{3}(x-\frac{5}{3})[/tex]
[tex]\Rightarrow -5y=2(x-\dfrac{5}{3})[/tex]
[tex]\Rightarrow -5y=2x-\dfrac{10}{3}[/tex]
[tex]\Rightarrow 2x+5y-\dfrac{10}{3}=0[/tex]
As it is given that all the coefficients are integers, so multiplying with 3
[tex]\Rightarrow 6x+15y-10=0[/tex]
Solution: As given line y =3x-5 meet x-axis at the point M.
On x axis y coordinate is zero.
Put y =0 in above equation, we get →x = 5/3
∴ Coordinate of M is (5/3,0).
As, also given , line 3y+2x=2 meets y-axis at point N.
On y axis , x coordinate is zero.
Substituting , x=0 in above equation, gives y =2/3.
Coordinate of point N is (0,2/3).
Equation of line passing through two points (a,b) and (p,q) is given by
→ [tex]\frac{y-b}{x-a} =\frac{q-b}{p-a}[/tex]
Or as X intercept = 5/3, and Y intercept = 2/3
Equation of line in intercept form is →[tex]\frac{x}{a} + \frac{y}{b} =1[/tex], where a and b is X intercept and y intercept respectively.
So, line passing through (5/3,0) and (0,2/3) is given by
→ [tex]\frac{x}{\frac{5}{3}} + \frac{y}{\frac{2}{3}}=1[/tex]
→ [tex]\frac{3x}{5} + \frac{3y}{2} =1[/tex]
→ 6 x + 15 y =10 [Taking LCM of 5 and 2 which is 10]
→ 6 x + 15 y -10=0, which is equation of the line joining M and N in the form ax + by + c = 0 where: a,b,c are integers.
