# Two lines both go through the point (1,2) one with slope 3 and one with slope 1/3. Find the...

## Question:

Two lines both go through the point {eq}(1,2) {/eq} one with slope {eq}3 {/eq} and one with slope {eq}1/3 {/eq}. Find the equations for the lines.

## Equation of a Line in Point-Slope Form

{eq}{/eq}

{eq}\text{Equation of a line that passes through a fixed point }(x_1 \ , \ y_1) \text{ and has slope m is given by } {/eq}

$$\displaystyle{\frac{y-y_1}{x-x_1} = m \\ or, \ y - y_1 = m(x-x_1) \\ }$$

Alternatively, we can write the equation as :

{eq}\displaystyle a(x-x_1) + b(y-y_1) = 0 \\ {/eq}

and use the relation

{eq}\displaystyle m = -\frac{a}{b} \\ {/eq}

{eq}{/eq}

Equation of a line in point-slope form is :

$$\displaystyle y - y_1 = m(x-x_1) \\$$

(1) When the line passes through (1, 2) and has slope 3

{eq}\displaystyle{y - 2 = 3(x-1) \\ \Rightarrow y = 3x - 1 \\ \Rightarrow 3x- y - 1 =0 \\ } {/eq}

(2) When the line passes through (1, 2) and has slope 1/3

{eq}\displaystyle{y - 2 = \frac{1}{3}(x-1) \\ \Rightarrow y = \frac{x}{3} + \frac{5}{3} \\ \Rightarrow x - 3y + 5 = 0 \\ } {/eq} 