# What is the point-slope form of the three lines that pass through (0,2), (4,5), and (0,0)?

Oct 4, 2016

The equations of three lines are $y = \frac{3}{4} x + 2$, $y = \frac{5}{4} x$ and $x = 0$.

#### Explanation:

The equation of line joining x_1,y_1) and x_2,y_2) is given by

$\frac{y - {y}_{1}}{{y}_{2} - {y}_{1}} = \frac{x - {x}_{1}}{{x}_{2} - {x}_{1}}$

while equation in pint slope form is of the type $y = m x + c$

Hence equation of line joining $\left(0 , 2\right)$ and $\left(4 , 5\right)$ is

$\frac{y - 2}{5 - 2} = \frac{x - 0}{4 - 0}$

or $\frac{y - 2}{3} = \frac{x}{4}$ or $4 y - 8 = 3 x$ or $4 y = 3 x + 8$ and

in point slope form it is $y = \frac{3}{4} x + 2$

and equation of line joining $\left(0 , 0\right)$ and $\left(4 , 5\right)$ is

$\frac{y - 0}{5 - 0} = \frac{x - 0}{4 - 0}$

or $\frac{y}{5} = \frac{x}{4}$ or $4 y = 5 x$ and

in point slope form it is $y = \frac{5}{4} x$

For equation of line joining $\left(0 , 0\right)$ and $\left(0 , 2\right)$, as ${x}_{2} - {x}_{1} = 0$ i.e. ${x}_{2} = {x}_{1}$, the denominator becomes zero and it is not possible to get equation. Similar would be the case if ${y}_{2} - {y}_{1} = 0$. In such cases as ordinates or abscissa are equal, we will have equations as $y = a$ or $x = b$.

Here, we have to find the equation of line joining $\left(0 , 0\right)$ and $\left(0 , 2\right)$. As we have common abscissa, the equation is

$x = 0$