Question

The line whose equation is 3x - 5y = 4 is dilated by a scale factor of centered at the origin. Whichstatement is correct?

Answer 1

Given:

The equation of a line(preimage) is,

`3x-5y=4`

Rewriting the above equation, we get

`\begin{gathered} -5y=-3x+4 \\ y=(3)/(5)x-(4)/(5)\text{ ------(1)} \end{gathered}`

The general equation of a straight line is,

`y=mx+c`

Here, m is the slope and c is the y intercept.

Comparing the above equations, we get that the slope of the line m=3/5 and the y intercept is -4/5.

The given line is dilated by a scale factor of 5/3 centered at the origin.

So, if (x,y) are the original coordinates of a point on the preimage of the line, then after dilation, the coordinates become (5x/3, 5y/3).

Replace (x,y) in equation (1) with (5x/3, 5y/3) to obtain the equation of the image of the line.

`\begin{gathered} (5y)/(3)=(3)/(5)((5x)/(3))-(4)/(5) \\ (5y)/(3)=x-(4)/(5) \\ y=(3)/(5)x-(12)/(25)\text{ ------(2)} \end{gathered}`

Comparing equations (1) and (2), we find that the slopes of the preimage and the image of the line has the same slope, but the y intercept changed.

Hence, the correct statement is "The image of the line has the same slope as the preimage but a different y-intercept."

Hence, option (a) is correct.