Respuesta :
Explanation
From the statement, we must find an equation:
0. of a line perpendicular to the line 5x - 4y = 6,
,1. that contains the point (x, y) = (-4, 3).
We write the equation of the given line as:
[tex]\begin{gathered} 4y+6=5x, \\ 4y=5x-6, \\ y=\frac{5}{4}x-\frac{6}{4}=\frac{5}{4}x-\frac{3}{2}. \end{gathered}[/tex]We see that the slope of the given line is m₁ = 5/4.
The slope-intercept equation of the line that we must find is:
[tex]y=m_2\cdot x+b_2.[/tex]Where:
• m₂ is the slope,
,• b₂ is the y-intercept.
1) From condition 1, we know that the line must be perpendicular to the given line. So their slopes must satisfy:
[tex]m_1\cdot m_2=-1\Rightarrow\frac{5}{4}\cdot m_2=-1.[/tex]Solving for m₂ we get:
[tex]m_2=-\frac{4}{5}.[/tex]2) Replacing the coordinates of the point from condition 2, and the slope m₂ = -4/5 in the equation of the line, we have:
[tex]\begin{gathered} 3=(-\frac{4}{5})\cdot(-4)+b_2, \\ 3=\frac{16}{5}+b_2. \end{gathered}[/tex]Solving for b₂ we get:
[tex]b_2=3-\frac{16}{5}=-\frac{1}{5}.[/tex]So the slope-intercept equation of the perpendicular line is:
[tex]y=-\frac{4}{5}x-\frac{1}{5}.[/tex]Answer[tex]y=-\frac{4}{5}x-\frac{1}{5}[/tex]