Discover the Algebraic Equation of a Line Parallel to y=-3/4x+2 Through (8,2)

Given the line parallel to the line

$y=-\frac{3}{4}x+2$

and passes through the point $(8,2)$.

Which of the algebraic representations is the corresponding one for the given line?

To solve this problem, we'll determine the equation of the line that is parallel to $y = -\frac{3}{4}x+2$ and passes through the point $(8,2)$.

Step 1: Identify the slope of the given line.
The slope ($m$) of the line $y = -\frac{3}{4}x + 2$ is $-\frac{3}{4}$, as it's the coefficient of $x$.

Step 2: Use the point-slope form, $y - y_1 = m(x - x_1)$, where $m = -\frac{3}{4}$ and the point $(x_1, y_1) = (8, 2)$.

Substitute into the point-slope form:
$y - 2 = -\frac{3}{4}(x - 8)$

Step 3: Simplify this equation to obtain the slope-intercept form:
$y - 2 = -\frac{3}{4}x + \frac{3}{4} \times 8$

Calculate the right side:
$y - 2 = -\frac{3}{4}x + 6$

Add 2 to both sides to isolate $y$:
$y = -\frac{3}{4}x + 6 + 2$
$y = -\frac{3}{4}x + 8$

This equation, $y = -\frac{3}{4}x + 8$, is in slope-intercept form and matches choice 4.

Thus, the equation of the line parallel to $y = -\frac{3}{4}x + 2$ and passing through $(8, 2)$ is $\boxed{y = -\frac{3}{4}x + 8}$.