Final answer:
To find the equation of a line perpendicular to 4x - 5y = 10 and passing through point P(5,6), we need to find the negative reciprocal of the slope of the given line. The slope-intercept form of the equation for the perpendicular line is y = -5/4x + 49/4.
Step-by-step explanation:
The equation of the line is given in the standard form as 4x - 5y = 10. To find the line that is perpendicular to this line, we need to find the negative reciprocal of the slope of the given line. The slope of the given line can be found by rewriting the equation in the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Rearranging the equation, we have 4x - 5y = 10 --> -5y = -4x + 10 --> y = 4/5x - 2.
Therefore, the slope of the given line is 4/5. The slope of a line perpendicular to this line will be the negative reciprocal, which is -5/4. Now, we have the slope of the perpendicular line and a point it passes through (P(5,6)). Using the point-slope form of a line, y - y1 = m(x - x1), we can now find the equation of the perpendicular line. Plugging in the values, we have y - 6 = -5/4(x - 5). Simplifying, we get y - 6 = -5/4x + 25/4. Rearranging the equation to the slope-intercept form, we get y = -5/4x + 25/4 + 6. Combining the terms, we have y = -5/4x + 49/4.
Therefore, the slope-intercept form of the equation of the line that is perpendicular to 4x - 5y = 10 and contains point P(5,6) is y = -5/4x + 49/4.