The final equation representing the new path that is perpendicular to the existing path and intersects it at the point (–3, 6) is : y = 1/3x + 7
To find the equation of the new path that will be perpendicular to the existing path, we need to follow a few steps.
1. Determine the slope of the existing path : The slope-intercept form of a line is given by y = mx + b, where m represents the slope and b represents the y-intercept. For the existing path, y = –3x – 3, we can see that the slope (m) is –3.
2. Find the slope of the perpendicular path : If two lines are perpendicular, their slopes are negative reciprocals of each other. The negative reciprocal of –3 is 1/3.
3. Use the point-slope form for the new path : The point-slope form of the equation of a line is given by (y - y1) = m(x - x1), where m is the slope and (x1, y1) is the point through which the line passes.
Our new path passes through the point (–3, 6) and has a slope of 1/3. Plugging these into point-slope form, we get :
y - 6 = 1/3(x - (-3))
y - 6 = 1/3(x + 3)
4. Put the equation into slope-intercept form : To put the above equation into slope-intercept form (y = mx + b), we need to simplify and solve for y.
y - 6 = 1/3(x) + 1/3(3)
y - 6 = 1/3(x) + 1
y = 1/3(x) + 1 + 6
y = 1/3(x) + 7, which is the equation that represents the new path.