graphing question that im not able to solve

Question

Find the standard form of the equation of the line through (4, 1) that is perpendicular to the line 2x = -3y + 7. What is the x- and y-intercept of the line

Answer 1

Find the slope, m from the equation below: $$2x=-3y+7$$ $$2x-7=-3y$$ $$-3y=2x-7$$ $$y=-\frac{2}{3}x+\frac{7}{3}$$ $$m=-\frac{2}{3}$$

To find the perpendicular slope, let: $$m_1=-\frac{2}{3}$$

Use the formula: $$m_1m_2=-1$$ $$\left(-\frac{2}{3}\right)m_2=-1$$ $$m_2=\frac{3}{2}$$

Use the basic formula: $$y=mx+c$$ $$y=\frac{3}{2}x+c$$

Substitute the coordinate $(4,1)$ to get $c$ value: $$y=\frac{3}{2}x+c$$ $$1=\frac{3}{2}\left(4\right)+c$$ $$1=6+c$$ $$c=-5$$

Therefore: $$y=\frac{3}{2}x+c$$ $$y=\frac{3}{2}x-5$$

Reference picture: