Respuesta :
We are asked that which line is perpendicular to the following equation.
[tex]3y+2x=12[/tex]First of all, we have to convert this equation into the slope-intercept form so that we can identify it's slope.
[tex]\begin{gathered} 3y+2x=12 \\ 3y=-2x+12 \\ y=\frac{-2x}{3}+\frac{12}{3} \\ y=\frac{-2x}{3}+4 \end{gathered}[/tex]Recall that the standard slope-intercept form is given by
[tex]y=mx+b[/tex]Where m is the slope and b is the y-intercept.
So comparing the standard form with the above equation, we find that
[tex]m_1=\frac{-2}{3}[/tex][tex]b=4[/tex]Now recall that the slopes of two perpendicular lines are negative reciprocals of each other.
[tex]m_1=-\frac{1}{m_2}[/tex]Therefore, the line perpendicular to the given equation will have a slope of
[tex]m_2=\frac{3}{2}[/tex]Finally, now we will check which given option has the exact above slope, that will be the correct equation.
Option A:
[tex]\begin{gathered} 6x-4y=24 \\ -4y=-6x+24 \\ y=\frac{6x}{4}-\frac{24}{4} \\ y=\frac{3}{2}x-6 \end{gathered}[/tex]This is the equation we were looking for since it has the slope m = 3/2
Therefore, the correct option is A.
The line 6x - 4y = 24 is perpendicular to the line 3y + 2x = 12
