Final answer:
To find the equation of a line that is perpendicular to the given line and passes through the point (2, 4), we need to find the negative reciprocal of the slope of the given line and use the point-slope form of the equation to find the equation of the perpendicular line. The equation of the line that is perpendicular to 5x - 2y = 8 and contains P(2, 4) is y = (-2/5)x + 24/5.
Step-by-step explanation:
To find the equation of a line that is perpendicular to another line, we need to find the negative reciprocal of the slope of the given line and use the given point to find the y-intercept.
- First, let's rearrange the equation 5x - 2y = 8 into the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
- Start by subtracting 5x from both sides, yielding -2y = -5x + 8.
- Then, divide both sides by -2 to isolate y and simplify the equation to y = (5/2)x - 4.
- Since we want the line that is perpendicular to this line, we need to find the negative reciprocal of the slope. The slope of the given line is 5/2, so the negative reciprocal is -2/5.
- Now we have the slope (-2/5) and the given point P(2, 4). We can substitute these values into the point-slope form of the equation, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
- Substituting x1 = 2, y1 = 4, and m = -2/5, we get y - 4 = (-2/5)(x - 2).
- Simplifying further, we have y - 4 = (-2/5)x + 4/5.
- Finally, rearrange the equation to get it in slope-intercept form. Add 4 to both sides to isolate y, resulting in y = (-2/5)x + 24/5.