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Find the equation of the straight line passing through the point (3,2) and perpendicular to the line
4 x - 3 y - 10

ANSWER IS 3x+4y-17=0​

Respuesta :

rspill6

Answer:

Unknown:  answer does not match given answer, but may be due to a missing term in the given line, as noted below.

Step-by-step explanation:

Let's look for an equation of the form y = mx + b, where m is the slope and b the y-intercept (the value of y when x is zero).

The best place to start is with the clue that the line we want is perpendicular to another line:  4x - 3y - 10.  But this expression, as written, is not a function since there is no "=" sign.  This assumes the correct expression is 4x - 3y - 10 = 0.  If not, the same steps may be used with the correct function.

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Rearrange 4x - 3y - 10 = 0 to point-slope format so that we can find the slope:

 4x - 3y - 10 = 0

  - 3y  = -4x + 10

      y = (4/3)x - (10/3)

The slope is (4/3).  The slope of any line perpendicular to this will have the negative inverse of (4/3), or - (3/4).  That means the line we are looking for will be of the form:

   y = -(4/3)x + b

The second clue is that this line intersects with point (3,2).  We need to find a value of b that will make this happen.  This is easy if we enter this point into the equation we have thus far and solve for b:

  y = -(4/3)x + b

  2 = -(4/3)*3 + b  for point (3,2)

 2 = -4 + b

 b = 6

 The line perpendicular to 4x - 3y - 10 = 0 and going through point (3,2) is

    y = -(4/3)x + 6

This doesn't match the given answer:

3x+4y-17=0

4y-17= -3x

4y= -3x + 17

 y = -(3/4)x + (17/4)

Please check the expression "4 x - 3 y - 10" and use the correct version in the same sequence as above.

semsee45

Answer:

[tex]3x+4y-17=0[/tex]

Step-by-step explanation:

To find the equation of the straight line passing through the point (3, 2) and perpendicular to the line 4x - 3y - 6 = 0, we first need to find the slope of the line 4x - 3y - 6 = 0 by rearranging the equation so that it is slope-intercept form, y = mx + b:

[tex]\begin{aligned}4x - 3y - 6 &= 0\\\\-4x+3y+6&=0\\\\3y&=4x-6\\\\y&=\dfrac{4}{3}x-2\end{aligned}[/tex]

Therefore, the slope of the given line is 4/3.

Since the slopes of perpendicular lines are negative reciprocals of each other, then the slope of the perpendicular line is:

[tex]m=-\dfrac{3}{4}[/tex]

Now, substitute the found slope m = -3/4 and the point (3, 2) into the point-slope formula to write the equation of the perpendicular line:

[tex]\begin{aligned}y-y_1&=m(x-x_1)\\\\y-2&=-\dfrac{3}{4}(x-3)\\\\4(y-2)&=-3(x-3)\\\\4y-8&=-3x+9\\\\3x+4y-8-9&=0\\\\3x+4y-17&=0\end{aligned}[/tex]

Therefore, the equation of the perpendicular line is:

[tex]\Large\boxed{\boxed{3x+4y-17=0}}[/tex]

[tex]\hrulefill[/tex]

Additional Notes

For the answer to be 3x + 4y - 17 = 0, then the original equation must be 4x - 3y - 6 = 0. Therefore, I have used 4x - 3y - 6 = 0 as the equation of the initial line in my answer.