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)(6) - color(blue)(2))/(color(red)(5) - color(blue)(3)) > 4/2 = 2
We can next use the point slope formula to write an equation for this 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)(2)(x - color(red)(3))
We can now transform this 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)(2) xx x) - (color(blue)(2) xx color(red)(3))
y - color(red)(2) = 2x - 6
First, we add color(blue)(2) and subtract color(red)(2x) from each side of the equation to isolate the x and y variables on the left side of the equation and the constant on the right side of the equation while keeping the equation balanced:
-color(red)(2x) + y - color(red)(2) + color(blue)(2) = -color(red)(2x) + 2x - 6 + color(blue)(2)
-2x + y - 0 = 0 - 4
-2x + y = -4
Now, we multiply each side of the equation by color(red)(-1) to ensure the coefficient for the x variable is positive while keeping the equation balanced:
color(red)(-1)(-2x + y) = color(red)(-1) xx -4
(color(red)(-1) xx -2x) + (color(red)(-1) xx y) = 4
color(red)(2)x - color(blue)(1)y = color(green)(4)
