# How do you write the equation given (-6,7); parallel to 3x + 7y = 3?

Jul 11, 2017

$s : y = - \frac{3}{7} x + \frac{31}{7}$

#### Explanation:

We take the line above in the form $r : y = a x + b$
We know that any parallel is $s : y = a x + c$
We choose $\left(- 6 , 7\right) \in s$

$7 y = 3 - 3 x \implies r : y = - \frac{3}{7} x + \frac{3}{7}$

$s : 7 = - \frac{3}{7} \left(- 6\right) + c$

$49 = 18 + 7 c \implies c = \frac{31}{7}$

Jul 11, 2017

$y = - \frac{3}{7} x + \frac{31}{7}$

$$         or


$7 y = - 3 x + 31$

#### Explanation:

Change $3 x + 7 y = 3$ to standard form of $y = m x + c$
$7 y = 3 - 3 x$
$y = - \frac{3}{7} x + \frac{3}{7}$

gradient, $m$, can be determined as $- \frac{3}{7}$

Two parallel lines would have the same gradient, in this case gradient of $- \frac{3}{7}$

You can choose to use gradient formula
Gradient, $m = \frac{{y}_{1} - {y}_{2}}{{x}_{1} - {x}_{2}}$
or general formula for straight line
$\left(y = m x + c\right)$

I would first be attempting it using gradient formula

replace $m$ with $- \frac{3}{7}$; replaace ${x}_{1} , {x}_{2} , {y}_{1} , {y}_{2}$ with $x$ and the x-coordinate, $y$ and the y-coordinate respectively

in this case, you have only 1 point given, if there is more, the x and y coordinate must be from the same point.

$- \frac{3}{7} = \frac{y - 7}{x - \left(- 6\right)}$
$- \frac{3}{7} = \frac{y - 7}{x + 6}$

rearrange
$- \frac{3}{7} \left(x + 6\right) = \frac{y - 7}{x + 6}$
$- \frac{3}{7} x - \frac{18}{7} = y - 7$
$y = - \frac{3}{7} x - \frac{18}{7} + 7$
$y = - \frac{3}{7} x + \frac{31}{7}$

in case you don't like fractions, multiply whole equation by 7
$7 y = - 3 x + 31$

Using general formula

replace $m$ with $- \frac{3}{7}$; $x$ with the x-coordinate, $y$ with the y-coordinate

in this case, you have only 1 point given, if there is more, the x and y coordinate must be from the same point.

$7 = \left(- \frac{3}{7}\right) \left(- 6\right) + c$

solve for c
$7 = \frac{18}{7} + c$
$c = \frac{31}{7}$

substitute $m$ and $c$ into $y = m x + c$
$y = - \frac{3}{7} x + \frac{31}{7}$
$7 y = - 3 x + 31$

Both equation for straight line you get are the same, depending on which you prefer.