We have the original line, which is represented as
[tex]4y=3x+7[/tex]Just to make things easier, let's write the same equation another way
[tex]4y-3x=7[/tex]That's the same equation! just rewrote.
We can easily find the transformed line using the fact that the image is always parallel to the ore-image (original equation), then, we must find the parallel equation!
We have
[tex]4y-3x=7[/tex]And we will look, which one is parallel to it? Simple! any equation that has
[tex]4y-3x\text{ or }3x-4y[/tex]On the left side, the number on the right side (without "x" and "y") doesn't matter.
Looking at our options, we can see that we have the following equation:
[tex]3x-4y[/tex]Then we can affirm that
[tex]3x-4y=9[/tex]Is parallel, see that if we multiply all by (-1) we get
[tex]\begin{gathered} 3x-4y=9 \\ 4y-3x=-9 \end{gathered}[/tex]The left side here is the same, then it's parallel, if it's parallel, it could represent its image.
____________________________ Second explanation
Let's use the slope-intercept form for the original equation, we have
[tex]4y=3x+7[/tex]If we divide both by 4 we get
[tex]y=\frac{3}{4}x+\frac{7}{4}[/tex]See that
[tex]m=\frac{3}{4}[/tex]Then we will look for a linear equation that has the same slope! if it has the same slope, it means they're parallel, then let's transform each equation in slope-intercept form.
To the right to the left we have
[tex]3x+4y=9\Rightarrow y=-\frac{3}{4}x+\frac{9}{4}[/tex]The slope here is m = -3/4, not 3/4, then it's not parallel.
[tex]4x+3y=9\Rightarrow y=-\frac{4}{3}x+\frac{9}{4}[/tex]Again, m is not 3/4, not parallel.
[tex]3x-4y=9\Rightarrow y=\frac{3}{4}x-\frac{9}{4}[/tex]Here the slope is 3/4, it's parallel to the original equation, then it's the correct answer 3x - 4y = 9.
And the last one
[tex]4x-3y=9\Rightarrow y=\frac{4}{3}x-\frac{9}{4}[/tex]The slope here is 4/3, wrong too
