Question

a line passes through point (-9, 7) and has a slope of \\(\frac{2}{3}\\). write an equation in \\(ax + by = c\\) form for this line. use integers for \\(a\\), \\(b\\), and \\(c\\).

Explanation:

Step1: Use point-slope form

The point-slope formula is $y - y_1 = m(x - x_1)$, where $(x_1,y_1)=(-9,7)$ and $m=\frac{2}{3}$.
$y - 7 = \frac{2}{3}(x - (-9))$

Step2: Simplify the right-hand side

Simplify the expression inside the parentheses first.
$y - 7 = \frac{2}{3}(x + 9)$
$y - 7 = \frac{2}{3}x + 6$

Step3: Eliminate the fraction

Multiply all terms by 3 to clear the denominator.
$3y - 21 = 2x + 18$

Step4: Rearrange to Ax+By=C form

Move all terms to one side to match the standard form, using integers.
$-2x + 3y = 39$
(Or multiply by -1 to make A positive: $2x - 3y = -39$, both are valid; the latter is often preferred with positive A)

Answer:

$2x - 3y = -39$ (or $-2x + 3y = 39$)