First, we need to determine the slope of the line. The slope can be found by using the formula: m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))
Where m is the slope and (color(blue)(x_1, y_1)) and (color(red)(x_2, y_2)) are the two points on the line.
Substituting the values from the points in the problem gives:
m = (color(red)(-5) - color(blue)(2))/(color(red)(3) - color(blue)(7)) = (-7)/-4 = 7/4
We can now use the point-slope formula to write an equation for the line. The point-slope formula states: (y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))
Where color(blue)(m) is the slope and (color(red)(x_1, y_1)) is a point the line passes through.
Substituting the slope we calculated and the values from the first point in the problem gives:
(y - color(red)(2)) = color(blue)(7/4)(x - color(red)(7))
We can now transform this equation to the Standard Form for a Linear Equation. The standard form of a linear equation is: color(red)(A)x + color(blue)(B)y = color(green)(C)
Where, if at all possible, color(red)(A), color(blue)(B), and color(green)(C)are integers, and A is non-negative, and, A, B, and C have no common factors other than 1
y - color(red)(2) = (color(blue)(7/4) xx x) - (color(blue)(7/4) xx color(red)(7))
y - color(red)(2) = 7/4x - 49/4
y - color(red)(2) + 2 = 7/4x - 49/4 + 2
y - 0 = 7/4x - 49/4 + (4/4 xx 2)
y = 7/4x - 49/4 + 8/4
y = 7/4x - 41/4
-color(red)(7/4x) + y = -color(red)(7/4x) + 7/4x - 41/4
-7/4x + y = 0 - 41/4
-7/4x + y = -41/4
color(red)(-4)(-7/4x + y) = color(red)(-4) xx -41/4
(color(red)(-4) xx -7/4x) + (color(red)(-4) xx y) = -cancel(color(red)(4)) xx -41/color(red)(cancel(color(black)(4)))
(-cancel(color(red)(-4)) xx -7/color(red)(cancel(color(black)(4)))x) - 4y = 41
color(red)(7)x - color(blue)(4)y = color(green)(41)
