Final answer:
The equation of the line that passes through the point (8, -4) and is parallel to the line x − 4y = 12 is y = ⅔x - 6. This is found by determining the slope ⅔ from the given line and applying it to the point-slope formula with the point (8, -4).
Step-by-step explanation:
To find an equation of the line that passes through the point (8, −4) and is parallel to the line given by x − 4y = 12, first, we need to find the slope of the given line. Rewriting the equation in slope-intercept form y = mx + b where m represents the slope, we have 4y = x - 12, which simplifies to y = ⅔x - 3. Therefore, our slope, m, is ⅔. Since parallel lines have the same slope, our line will also have a slope of ⅔.
Next, we use the point-slope form (y - y₁) = m(x - x₁) where (x₁, y₁) is a point on the line and m is the slope. Substituting the given point (8, −4) and the slope ⅔, we get y + 4 = ⅔(x - 8). To convert this into slope-intercept form, we distribute ⅔ and simplify: y + 4 = ⅔x - 2, so y = ⅔x - 6.
Therefore, the equation of the line that is parallel to x − 4y = 12 and passes through the point (8, −4) is y = ⅔x - 6.