Answer:
y -6 = -7/2(x -4)
Step-by-step explanation:
You want the line parallel to y = -7/2x +9 that passes through the point of intersection of x+y=10 and 3x-4y+12=0.
Point of intersection
First of all, we must find the point of intersection of the two given lines. This is done easily using a graphing application (see attached).
We can also use "elimination" to find the point of intersection. Adding 4 times the first equation to the second, we get ...
4(x +y) +(3x -4y +12) = 4(10) +(0)
7x +12 = 40 . . . . . . . simplify
x = (40 -12)/7 = 4 . . . subtract 12 and divide by 7
y = 10 -x = 6
The point of intersection is (x, y) = (4, 6).
Point-slope equation
A line parallel to the given line will have the same slope. That slope is the x-coefficient in the slope-intercept equation y = -7/2x +9. That is, the slope of the desired line will be -7/2.
The point-slope equation for a line is ...
y -k = m(x -h) . . . . . . . line with slope m through point (h, k)
Using the point and slope for the desired line, we see its equation is ...
y -6 = -7/2(x -4) . . . . . . line with slope -7/2 through point (4, 6)
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Additional comment
Other forms of the equation of the line are ...
y = -7/2x +20 . . . . . . slope-intercept form
7x +2y = 40 . . . . . . . standard form
