The y-intercept of #7# is equal to the point #(0, 7)#. Using this point and the point from the problem we can determine the slope. 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)(7) - color(blue)(2))/(color(red)(0) - color(blue)(5)) = 5/-5 = -1#
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 y-intercept of #(0, 7)# gives:
#(y - color(red)(7)) = color(blue)(-1)(x - color(red)(0))#
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 above to the standard form as follows:
#(y - color(red)(7)) = color(blue)(-1)(x - color(red)(0))#
#y - color(red)(7) = -1x - (-1 xx color(red)(0))#
#y - color(red)(7) = -1x - 0#
#y - color(red)(7) = -1x#
#color(blue)(1x) + y - color(red)(7) + 7 = color(blue)(1x) - 1x + 7#
#1x + y - 0 = 0 + 7#
#color(red)(1)x + color(blue)(1)y = color(green)(7)#
