a line passes through (2,4) and (-2,2). find the value of y if (6,y) lies on the same line

Accepted Solution

Answer:y = 6, rendering the coordinate pair: (6,6)Step-by-step explanation:We start by writing the equation of the line that passes through two given points on the plane: [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] beginning with finding the slope of the segment that joints the points using the slope formula: [tex]slope=\frac{y_2-y_1}{x_2-x_1}[/tex] Let's call [tex](x_1,y_1)[/tex] = (2,4), and [tex](x_2,y_2)[/tex] = (-2,2). Then we have the formula for the slope: [tex]slope=\frac{y_2-y_1}{x_2-x_1}=\frac{2-4}{-2-2} =\frac{-2}{-4} =\frac{1}{2}[/tex]Now that we have the slope of the line, we can find the actual equation of the line by using one of the given points, and the "point-slope" form of a line with slope "m" and going through a point [tex](x_0,y_0)[/tex] - which in our case we defie as one of our given points, let's say (2, 4):[tex]y-y_0=m\,(x-x_0)\\y-4=\frac{1}{2} (x-2)\\y-4=\frac{1}{2} x-1\\y=\frac{1}{2} x-1+4\\y=\frac{1}{2} x+3[/tex]now we find what is the "y" value in such line that corresponds to an x-value of "6" to complete the coordinate pair (6, ?). For such we simply evaluate the equation above at x = 6:[tex]y=\frac{1}{2} x+3\\y=\frac{1}{2} (6)+3\\y=3+3\\y=6[/tex]Therefore, y must be "6".