Find the Line Equation: Parallel to 5y-5x=15 Through Point (-5,12)

To solve this problem, we'll follow these steps:

• Step 1: Determine the slope of the given line $5y - 5x = 15$.
• Step 2: Use this slope and the point $(-5, 12)$ in the point-slope form.
• Step 3: Rearrange the equation to find the required line's equation.

Let's work through each step:

Step 1: Determine the slope of the given line.
First, convert the equation $5y - 5x = 15$ to slope-intercept form ($y = mx + b$):

Divide every term by 5 to simplify:
$5y = 5x + 15$
$y = x + 3$

The slope ($m$) of this line is $1$.

Step 2: Use the point-slope form with the given point and slope.
We have a point $(-5, 12)$ and a slope $m = 1$. The point-slope form is:

$y - y_1 = m(x - x_1)$
Substitute $y_1 = 12$, $x_1 = -5$, and $m = 1$:
$y - 12 = 1(x + 5)$

Simplify the equation:
$y - 12 = x + 5$

Step 3: Rearrange to get the solution.
Rearrange the equation:
$y = x + 5 + 12$
$y = x + 17$

We want to express this in the form of one of the given choices:
$y - x = 17$.

Therefore, the solution to the problem is $y - x = 17$.

This matches the final answer choice given in the problem, confirming our solution is correct.