Question

Find the slope and y intercept of √3x+(1-√3)y= 3

Answer 1

Answer:

slope, $m=\frac{-\sqrt{3}}{1-\sqrt{3}}$

y-intercept, $c=\frac{3}{1-\sqrt{3}}$

Step-by-step explanation:

Find the slope and y intercept of

√3x+(1-√3)y= 3

Equation of given line is

$\sqrt{3}x+(1-\sqrt{3})y=3$

$\implies\:(1-\sqrt{3})y=-\sqrt{3}x+3$

$\implies\:y=\frac{-\sqrt{3}}{1-\sqrt{3}}x+\frac{3}{1-\sqrt{3}}$

comparing this with

$\boxed{\bf{y=mx+c}}$

we get

slope, $m=\frac{-\sqrt{3}}{1-\sqrt{3}}$

y-intercept, $c=\frac{3}{1-\sqrt{3}}$

Answer 2

• Slope is $\sf\dfrac{(3\:+\:{\sqrt{3}}}{2})$ and
• y intercept - $\sf\dfrac{(3\:+\:{\sqrt{3}}}{2})$

Step-by-step explanation:

Given,

• Equation = $\sf{x\:{ \sqrt{3}}\:+\:y(1\:-\:{ \sqrt{3}}\:=\:3}$

$\sf{y\:(1\:-\:{ \sqrt{3}})\:=\:-x\:{ \sqrt{3}}\:+\:3}$

Dividing by $\sf{(1\:-\:{ \sqrt{3}}}),$

⟼ $\sf{y\:=\:({ \dfrac{-x{ \sqrt{3}}}{1\:-\:{ \sqrt{3}}}}\:+\:{ \dfrac{3}{1\:-\:{ \sqrt{3}}}}})$

⟼ $\sf{x({ \dfrac{{ {-}\sqrt{3}}}{1\:-\:{ \sqrt{3}}}}})\:+\:{\dfrac{3}{1\:-\:{ \sqrt{3}}}}$

⟼ $\sf{y\:=\:x({ \dfrac{{ \sqrt{3}}}{{ \sqrt{3}\:-\:1}}}})\:+\:{\dfrac{3}{1\:-\:{ \sqrt{3}}}}\:----(1)$

Comparing (1) with 'y = mx + c',

⟼ $\sf{m\:=}$ $\sf\dfrac{{ \sqrt{3}}}{{ \sqrt{3}}\:-\:1}$ $\sf{and\:c\:=}$$\sf\dfrac{3}{1\:-\:{\sqrt{3}}}$

⟼ $\sf{slope\:=\:m\:=}$ $\sf\dfrac{{\sqrt{3}}}{{\sqrt{3}}\:-\:1}\:=$ $\sf\dfrac{{\sqrt{3}}}{{\sqrt{3}}\:-\:1}\:×$ $\sf\dfrac{{\sqrt{3}}\:+\:1}{{\sqrt{3}}\:+\:1}$

Rationalizing the denominator,

⟼ $\sf{m\:=}$ $\sf\dfrac{{\sqrt{3}}\:{\sqrt{3}}\:+\:1}{{\sqrt{3}}\:-\:1}\:{\sqrt{3}}\:+\:1$

⟼ $\sf{m\:=}$ $\sf\dfrac{3\:+\:{\sqrt{3}}}{3\:-\:1}$

⟼ $\sf{m\:=}$ $\sf\dfrac{3\:+\:{\sqrt{3}}}{2}\:=$ ] $\sf\dfrac{3\:+\:{\sqrt{3}}}{2}\:=$

⟼ $\sf{and\:c\:=}$ $\sf\dfrac{3}{1\:-\:{\sqrt{3}}}\:×$ $\sf\dfrac{1\:+\:{\sqrt{3}}}{1\:+\:{\sqrt{3}}}$

Rationalizing the denominator,

⟼ $\sf{c\:=}$ $\sf\dfrac{3\:(1\:+\:{\sqrt{3}})}{1\:-\:3}$

⟼ $\sf{c\:=}$ $\sf\dfrac{3\:+\:{\sqrt{3}}}{-2}\:=\:-$ $\sf\dfrac{(3\:+\:{\sqrt{3}}}{2})$

• Hence, slope is $\sf\dfrac{(3\:+\:{\sqrt{3}}}{2})$ and y intercept - $\sf\dfrac{(3\:+\:{\sqrt{3}}}{2})$.