# What is the equation of the line passing through (3,-34) and (4,-9)?

May 2, 2016

The line is: $y = 25 x - 109$

#### Explanation:

There are different methods to approach this:
$1.$. Form simultaneous equations based on $y = m x + c$
(Substitute the values of $x \mathmr{and} y$ which have been given.)

$- 34 = m \left(3\right) + c$ and $- 9 = m \left(4\right) + c$

Solve them to find the values of $m \mathmr{and} c$, which will give the equation of the line. Elimination by subtracting the 2 equations is probably the easiest as the $c$ terms will subtract to 0.

$2.$ Use the two points to find the gradient. $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
Then substitute values for $m$ and one point $x , y$ into $y = m x + c$ to find $c$.
Finally answer in the form $y = m x + c$, using the values for $m \mathmr{and} c$ you have found.

$3.$ Use the formula from coordinate (or analytical) geometry which uses 2 points and a general point $\left(x , y\right)$
$\frac{y - {y}_{1}}{x - {x}_{1}} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

Substitute the values for the 2 given points, calculate the fraction on the right hand side (which gives the gradient), cross-multiply and with a small amount of transposing, the equation of the line is obtained.

$\frac{y - \left(- 34\right)}{x - 3} = \frac{- 9 - \left(- 34\right)}{4 - 3} = \frac{25}{1}$

$\frac{y + 34}{x - 3} = \frac{25}{1}$ Now cross-multiply

$y + 34 = 25 x - 75$

$y = 25 x - 109$