Final answer:
The equation of the line perpendicular to the given line and passing through (2,3) is y = -5/4x + 13/2. This is found by first determining the negative reciprocal of the original line's slope and then using the point-slope form of the equation with the given point.
Step-by-step explanation:
To find the equation of a line that is perpendicular to the given line 4x - 5x = 7 and passes through the point (2,3), we first need to correct the given equation, as it appears there is a typo. Assuming the equation should have different coefficients for x, let's consider the example of a line with the form 4x - 5y = 7. We begin by finding the slope of this line. Rewriting the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, we have 5y = 4x - 7, and then y = (4/5)x - 7/5. Hence, the slope (m) of the original line is 4/5.
A line that is perpendicular to another line has a slope that is the negative reciprocal of the original line's slope. Therefore, our new line has a slope of -5/4. Using the point-slope form of a line equation, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point through which the line passes, we substitute the known point (2, 3) and the slope -5/4 to find the equation of our line: y - 3 = -5/4(x - 2).
To write this equation in slope-intercept form, we would distribute the slope across the parenthesis and add 3 to both sides to find the y-intercept. The final equation is y = -5/4x + 13/2, which represents the line perpendicular to the original line and passing through the point (2,3).