The slope-intercept form:
[tex]y=mx+b[/tex]
m - slope
b - y-intercept
Convert 2x + 5y = 10 to the slope0intercept form:
[tex]2x+5y=10[/tex] subtract 2x from both sides
[tex]5y=-2x+10[/tex] divide both sides by 5
[tex]y=-\dfrac{2}{5}x+2[/tex]
Let [tex]k:y=m_1x+b_1[/tex] and [tex]l:y=m_2x+b_2[/tex]
[tex]l\ \perp\ k\iff m_1m_2=-1\to m_2=-\dfrac{1}{m_1}[/tex]
We have [tex]m_1=-\dfrac{2}{5}[/tex]
Therefore
[tex]m_2=-\dfrac{-\frac{2}{5}}=\dfac{5}{2}[/tex]
We have the equation of a line:
[tex]y=\dfrac{5}{2}x+b[/tex]
Put the coordinates of the point (5, 1) to the equation of a line:
[tex]1=\dfrac{5}{2}(5)+b[/tex]
[tex]1=\dfrac{25}{2}+b[/tex] subtract [tex]\dfrac{25}{2}[/tex] from both sides
[tex]-\dfrac{23}{2}=b\to b=\dfrac{23}{2}[/tex]
Answer: [tex]\boxed{y=\dfrac{5}{2}x+\dfrac{23}{2}}[/tex]
