Final answer:
To find the line perpendicular to x + 4y = 24 and passing through the point (5,-2), we need to determine the slope of the given line and find its negative reciprocal. Then we can use the point-slope form of an equation to find the y-intercept.
Step-by-step explanation:
To find the line that is perpendicular to the line x + 4y = 24 and passes through the point (5,-2), we need to determine the slope of the given line and find its negative reciprocal.
The slope of the given line can be found by rearranging the equation in the form y = mx + b, where m is the slope. The equation becomes y = (-1/4)x + 6. Therefore, the slope is -1/4.
The negative reciprocal of -1/4 is 4/1, which means the perpendicular line will have a slope of 4.
Now, we can plug the point (5,-2) and the new slope into the equation y = mx + b to find the y-intercept, b. The equation becomes -2 = 4(5) + b, which simplifies to -2 = 20 + b. Solving for b, we get b = -22.
Thus, the equation of the line perpendicular to x + 4y = 24 and passing through the point (5,-2) is 4x - y = -22.