Respuesta :
The perpendicular line to any other line will have a slope that is the negative reciprocal of the other slope. That means flipping the fraction representing the slope and changing its sign.
To find the slope of a line perpendicular to 6x - 4y = 3 it will be convenient to express this equation in the slope-intercept form to see more clearly the value of its slope. For that we simply isolate the y:
[tex]\begin{gathered} 6x-4y=3 \\ 6x-3=4y \\ y=\frac{6}{4}x-\frac{3}{4} \\ y=\frac{3}{2}x-\frac{4}{3} \end{gathered}[/tex]We can clearly see now the slope of the first line:
[tex]\frac{3}{2}[/tex]The slope of the given line 6x -4y = 3 is 3/2.
We know now that the slope of the perpendicular line is the negative reciprocal of 3/2, or well:
[tex]-\frac{2}{3}[/tex]Then, the slope of the line that is perpendicular to the above line is -2/3
Now we can begin building the equation. Recalling the slope-intercept form of the equation of a line:
[tex]y=mx+b[/tex]Where m is the slope of the line, and b is the y-intercept. We know then the value of m:
[tex]y=-\frac{2}{3}x+b[/tex]We know that the line passes through the point (6, -1). We can use this information to find the value of b and finish building the equation completely.
We can just find the value of b such that when we replace the x and y-values of the point, the equation is satisfied:
[tex]\begin{gathered} -1=-\frac{2}{3}(6)+b \\ b=-1+\frac{2}{3}\cdot6 \\ b=-1+\frac{12}{3} \\ b=-1+4 \\ b=3 \end{gathered}[/tex]Now knowing the y-intercept we can finally find the equation:
[tex]y=-\frac{2}{3}x+3[/tex]The equation of the perpendicular line that passes through (6, -1) is y = -(2/3)x + 3
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