Question

Write an equation in the variables x and y for the line that passes through the point (4, 7) and that is perpendicular to the line 2x + 2y = 14. provide precise expressions, not decimal approximations.

Answer 1

First of all, we need to find the slope of the given line, writing it in the $y=mx+q$ form and reading the $m$ coefficient:

$2x + 2y = 14 \iff 2y = -2x + 14 \iff y = -x+7$

So, the slope is $-1$

If a line with slope $m'$ is perpendicular to a line with slope $m$, the following relationship holds:

$mm' = -1$

So, the slope of our line must satisfy

$m(-1) = -1 \iff m=1$

So, we need the equation of a line with slope 1 and passing through (4,7). Given the slope $m$ and a point $(x_0,y_0)$ of a given line, its equation is given by

$y-y_0 = m(x-x_0)$

So, if you plug your values, you have

$y-7=1\cdot(x-4) \iff y = x+3$