How do you write the equation of the line passing through the points (3,4) and (1,-6)?

Jun 24, 2016

$y = 5 x - 11$

Explanation:

There are several methods to choose from:

Method 1. Use the given points to find the gradient, then substitute one of the points as (x,y) as well as the slope (m), into $y = m x + c$ to find the value of $c$.
Then substitute the values of m and c to find the final equation.

Method 2. Make simultaneous equations using each point as an (x,y) but with m and c as the unknowns. Solve for m and c and substitute them into $y = m x + c$

Method 3. There is a neat formula for exactly this case. "Find the equation if you are given two points". Quick and easy!

$\frac{y - {y}_{1}}{x - {x}_{1}} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} \text{ for } \left(3 , 4\right) \mathmr{and} \left(1 , - 6\right)$

$\frac{y - \left(- 6\right)}{x - 1} = \frac{4 - \left(- 6\right)}{3 - 1} = \frac{10}{2} = \frac{5}{1}$

$\frac{y + 6}{x - 1} = \frac{5}{1} \text{ cross multiply and re-arrange}$

$y + 6 = 5 x - 5$

$y = 5 x - 5 - 6$
$y = 5 x - 11$