To find the equation of a line that passes through the point (10, 3) and is perpendicular to the line described by the equation y = 5x + 7, we first need to determine the slope of the original line.
The given line has a slope of 5. Two lines are said to be perpendicular if the product of their slopes is -1.
To find the slope of the perpendicular line, we can use the formula:
m1 × m2 = -1
Here, m1 is the slope of the original line (5), and m2 is the slope of the line we are trying to find. By rearranging the formula, we find:
m2 = -1/m1 = -1/5
Now that we have the slope of the new line, which is -1/5, we can use the point-slope form of the equation of a line to find the equation:
y – y1 = m(x – x1)
Here, (x1, y1) is the point (10, 3) through which the line passes, and m = -1/5.
Substituting the known values into the point-slope formula gives us:
y – 3 = -1/5(x – 10)
To express this equation in slope-intercept form (y = mx + b), we simplify:
First, distribute the slope:
y – 3 = -1/5x + 2
Then, add 3 to both sides:
y = -1/5x + 5
Therefore, the equation of the line that passes through the point (10, 3) and is perpendicular to the line y = 5x + 7 is:
y = -1/5x + 5