The slope-intercept form:
[tex]y=mx+b[/tex]
m - slope
b - y-intercept
Convert the equation of the line 4x - 7y = 7x + 4y to the slope-intercept form:
[tex]4x-7y=7x+4y[/tex] subtract 4x from both sides
[tex]-7y=3x+4y[/tex] subtract 4y from both sides
[tex]-11y=3x[/tex] divide both sides by (-11)
[tex]y=-\dfrac{3}{11}x[/tex]
Let [tex]k:y=m_1x+b_1[/tex] and [tex]l:y=m_2x+b_2[/tex] then
[tex]l\ \perp\ k\iff m_1m_2=-1\to m_2=-\dfrac{1}{m_1}[/tex]
We have [tex]m_1=-\dfrac{3}{11}[/tex] therefore the slope of the line perpendicular is
[tex]m_2=-\dfrac{1}{-\frac{3}{11}}=\dfrac{11}{3}[/tex]
Therefore we have:
[tex]y=\dfrac{11}{3}x+b[/tex]
Put the coordinates of the point (4, -4) to the equation of a line:
[tex]-4=\dfrac{11}{3}(4)+b[/tex]
[tex]-4=\dfrac{44}{3}+b[/tex]
[tex]-\dfrac{12}{3}=\dfrac{44}{3}+b[/tex] subtract [tex]\dfrac{44}{3}[/tex] from both sides
[tex]-\dfrac{56}{3}=b[/tex]
Answer: [tex]\boxed{y=\dfrac{11}{3}x-\dfrac{56}{3}}[/tex]
