line l passes through the points 1,6 and -2,-9. write an equation of the image of l after a dilation with a scale factor of 5 centered at the origin

Answer:Equation of image I is, y = 5x + 5Step-by-step explanation:An Equation of line passing through the two points [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] is given by;[tex]y-y_1 = m(x-x_1)[/tex] where m is the slope of the line.Given: Line I passes through the points (1, 6)  and (-2, -9)To find an equation of the image of I after a dilation of scale factor 5 centered at origin.Dilation: A  transformation in which a image grows larger. It may be with respect to a point or with respect to the axis of a graph.Since, dilation requires a center point and a scale factor k.The rule of dilation with a scale factor k =5 centered at origin is given by:[tex](x, y) \rightarrow (5x , 5y)[/tex]Now, to dilate the points of I are;[tex](1, 6) \rightarrow (5 \cdot 1 , 5 \cdot 6)[/tex] = (5 , 30)[tex](-2, -9) \rightarrow (5 \cdot -2 , 5 \cdot -9)[/tex] = (-10 , -45)The points of image I are; (5, 30) and (-10 , -30)First calculate the slope:Slope(m) of the Image I is given by:[tex]m = \frac{y_2-y_1}{x_2-x_1}[/tex]  then;[tex]m = \frac{-45-30}{-10-5} =\frac{-75}{-15} = 5[/tex]Then, the equation of image I is;[tex]y-30 = 5(x-5)[/tex]Using distributive property;  [tex]a \cdot (b+c) = a\cdot b + a\cdot c[/tex]y -30 =5x -25Add 30 to both sides we get;y -30+30 = 5x -25 +30Simplify:y = 5x + 5The equation of the image I after a dilation with scale factor of 5 centered at the origin is, y = 5x + 5