Respuesta :

For lines to be perpendicular, the slope of one of them must be the inverse negative of the other one. So for example given the lines:

[tex]y=mx+b[/tex][tex]y=nx+c[/tex]

For them to be perpendicular their slopes must be:

[tex]n=-\frac{1}{m}[/tex]

To determine the equation of a line perpendicular to the given one, the first step is to calculate the slope of the graphed line.

Using two points of the line, for example: (0,3) and (4,6)

[tex]\begin{gathered} m=\frac{y_1-y_2}{x_1-x_2} \\ m=\frac{6-3}{4-0}=\frac{3}{4} \end{gathered}[/tex]

The slope of the graphed line is

[tex]m=\frac{3}{4}[/tex]

Now the slope of the perpendicular line must be the inverse negative so that:

[tex]\begin{gathered} n=-\frac{1}{m} \\ n=-\frac{4}{3} \end{gathered}[/tex]

Now that we know the slope of the perpendicular line, and is given that it crosses the point (3,-1) we can use the point slope form to determine its equation

[tex]y-y_1=m(x-x_1)[/tex]

For our line

[tex]\begin{gathered} y-(-1)=-\frac{4}{3}(x-3) \\ y+1=-\frac{4}{3}x+4 \\ y=-\frac{4}{3}x+4-1 \\ y=-\frac{4}{3}x+3 \end{gathered}[/tex]

The equation of a perpendicular line of the one shown in the graph rthat goes through point (3,-1) is

[tex]y=-\frac{4}{3}x+3[/tex]

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