Question

Find the equation of the line that is perpendicular to the line 6x-4y=3 and passes through the point (6,-1) Find the slope of the given line 6x-4y=3 what is the slope of a line that is perpendicular to the above line? Find the equation of the perpendicular line that passes through (6,-1) Step By Step Explanation please.

Answer 1

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:

`\begin{gathered} 6x-4y=3 \\ 6x-3=4y \\ y=(6)/(4)x-(3)/(4) \\ y=(3)/(2)x-(4)/(3) \end{gathered}`

We can clearly see now the slope of the first line:

`(3)/(2)`

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:

`-(2)/(3)`

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:

`y=mx+b`

Where m is the slope of the line, and b is the y-intercept. We know then the value of m:

`y=-(2)/(3)x+b`

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:

`\begin{gathered} -1=-(2)/(3)(6)+b \\ b=-1+(2)/(3)\cdot6 \\ b=-1+(12)/(3) \\ b=-1+4 \\ b=3 \end{gathered}`

Now knowing the y-intercept we can finally find the equation:

`y=-(2)/(3)x+3`

The equation of the perpendicular line that passes through (6, -1) is y = -(2/3)x + 3

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