First, we need to determine the slope of the line passing through the two points. 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)(3) - color(blue)(2))/(color(red)(6) - color(blue)(2)) = 1/4#
Next, we can use the point-slope formula to find 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 first point from the problem gives:
#(y - color(red)(2)) = color(blue)(1/4)(x - color(red)(2))#
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. We can convert the equation in point-slope form to standard form as follows:
First, we need to remove the fraction as all coefficients must be integers. We will multiply each side of the equation by #color(red)(4)#:
#color(red)(4)(y - 2) = color(red)(4) xx 1/4(x - 2)#
#(color(red)(4) xx y) - (color(red)(4) xx 2) = cancel(color(red)(4)) xx 1/color(red)(cancel(color(black)(4)))(x - 2)#
#4y - 8 = 1x - 2#
Next, we will add #color(red)(8)# and subtract #color(blue)(1x)# from each side of the equation to isolate the #x# and #y# terms on the left side of the equation:
#-color(blue)(1x) + 4y - 8 + color(red)(8) = -color(blue)(1x) + 1x - 2 + color(red)(8)#
#-1x + 4y - 0 = 0 + 6#
#-1x + 4y = 6#
Now, we will multiply each side of the equation by #color(red)(-1)# to make the #x# coefficient a positive integer:
#color(red)(-1)(-1x + 4y) = color(red)(-1) xx 6#
#color(red)(1)x + color(blue)(-4)y = color(green)(-6)#
