Answer:
12x -10y -235 = 0 . . . . in general form; has integer coefficients
Step-by-step explanation:
You want the line through points (18, -1.9) and (30, 12.5).
It can be helpful to look at the difference vector (∆x, ∆y). Here, that is ...
(x2 -x1, y2 -y1) = (30 -18, 12.5 -(-1.9)) = (12, 14.4)
The vector in reduced integer form is ...
(12, 14.4) = (1, 1.2) = (5, 6)
Given this direction vector, there are a number of ways the equation of the line can be written. One of them is ...
(∆y)(x -x1) -(∆x)(y -y1) = 0 . . . . . for some point (x1, y1)
Using the first of the given points, this would be ...
6(x -18) -5(y -(-1.9)) = 0
6x -5y -117.5 = 0
Multiplying by 2 gives the general form equation ...
12x -10y -235 = 0
The equation was written in general form: ax +by +c = 0.
This is readily converted to standard form: ax +by = c.
We recognize from the non-integer point values that it is likely the usual slope-intercept or point-slope forms would involve mixed numbers and/or fractions. We like integers, so chose a form that makes use of integer coefficients.
As we saw in the development above, obtaining the final integer form required multiplying by a factor (2) that eliminates the decimal fraction. In general, for a 1 decimal place fraction, that multiplier will be 2 or 10.
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Additional comment
The slope is ∆y/∆x = 6/5 = 1.2, and the y-intercept is 235/-10 = -23.5. That means the slope-intercept form is ...
y = 1.2x -23.5
Decimal coefficients are suitable for expression in this form.
Another "direction vector" form is ...
(x -x1)/∆x = (y -y1)/∆y
If you subtract the right side from both sides and multiply by (∆x)(∆y), you get the form we used above.
If you define the parameter t = (x -x1)/∆x, then the above equation can be written in parametric form as ...
(x, y) = (t·∆x +x1, t·∆y +y1)
