Final answer:
The equation of the line parallel to 3y + 24x = 9 and going through the point (8, 5) is y = -8x + 69. This was determined by finding the slope of the original line, which is -8, and then using the point-slope form with the given point to find the new line's equation.
Step-by-step explanation:
To find an equation for a line that is parallel to a given line and passes through a specific point, one must first identify the slope of the given line. The standard form of the line provided is 3y + 24x = 9. To find the slope, this equation can be rewritten in slope-intercept form, which is y = mx + b, with m representing the slope and b the y-intercept. First we solve for y:
- 3y = -24x + 9
- y = -24x/3 + 9/3
- y = -8x + 3
The slope of the given line is -8. Since parallel lines have the same slope, the new line will also have a slope of -8. Now, using the point that the new line must pass through, which is (8, 5), we apply the point-slope form of the equation: y - y1 = m(x - x1), where (x1, y1) are the coordinates of the given point and m is the slope. Substituting our values, we get:
- y - 5 = -8(x - 8)
- y - 5 = -8x + 64
- y = -8x + 64 + 5
- y = -8x + 69
The equation of the line parallel to 3y + 24x = 9 and passing through the point (8, 5) is thus in the y = mx + b form:
y = -8x + 69