# How do you find the value of k if the line through the points (2k-2,27-k) and (5k-5,3k+4) is parallel to the line through the point (15,7) and (-9,-6)?

Jun 20, 2018

If they are parallel then the gradients are the same

$\frac{3 k + 4 - \left(27 - k\right)}{5 k - 5 - \left(2 k - 2\right)} = \frac{7 - - 6}{15 - - 9}$

Remove the brackets

$\frac{3 k + 4 - 27 + k}{5 k - 5 - 2 k + 2} = \frac{7 + 6}{15 + 9}$

Collect like terms

$\frac{4 k - 23}{3 k - 3} = \frac{13}{24}$

Multiply $\left(4 k - 23\right)$ by 24
Multiply 13 by $\left(3 k - 3\right)$

$96 k - 552 = 39 k - 39$

Subtract $39 k$

$57 k - 552 = - 39$

$k = 9$