Final answer:
To find the equation of a line perpendicular to -9x-4y = -38 that passes through (2,5), we first find the slope of the given line, which is -9/4. Then, we use the negative reciprocal of that slope, which is 4/9, in the point-slope form along with the given point to get y = (4/9)x + (37/9) as the final equation.
Step-by-step explanation:
The student is asking for the equation of a line that is perpendicular to the given line -9x-4y = -38 and passes through the point (2,5). To find the equation of the perpendicular line, we must first put the given line in slope-intercept form (y = mx + b) to determine its slope. The negative reciprocal of this slope will be the slope of the perpendicular line.
The equation -9x-4y = -38 can be rewritten as 4y = -9x + 38, and then y = (-9/4)x + (38/4). The slope of this line is -9/4, so the slope of the perpendicular line is the negative reciprocal, which is 4/9. Using the point-slope form of a line (y - y1 = m(x - x1)), where m is the slope and (x1, y1) is the point the line goes through, we plug in the slope 4/9 and the point (2,5).
Applying the point-slope form, we get y - 5 = (4/9)(x - 2). To put this into slope-intercept form, we distribute and simplify: y - 5 = (4/9)x - (4/9)×2, which simplifies to y - 5 = (4/9)x - 8/9. Adding 5 to both sides gives us y = (4/9)x + (45/9 - 8/9), and finally, y = (4/9)x + (37/9). This is the equation of the perpendicular line that passes through the point (2,5).