Question

Which equations represent the line that is perpendicular to the line 5x − 2y = −6 and passes through the point (5, −4)? Select three options. y = –Two-fifthsx – 2 2x + 5y = −10 2x − 5y = −10 y + 4 = –Two-fifths(x – 5) y – 4 = Five-halves(x + 5)

Answer 1

Answer:

Option B) is the correct equation.

Step-by-step explanation:

We are given the following information in the question:

We are given a line:

$5x -2y = -6\\-2y = -6-5x\\2y = 5x + 6\\\\y = \displaystyle\frac{5}{2}x + 3$

Comparing the given equation with the general equation of line:

$y = mx + c\\\text{where m is the slope of line and c is the y-intercept}$

We get,

$m_1 = \displaystyle\frac{5}{2}, c_1 = 3$

When two lines are perpendicular, then the slopes of lines satisfy:

$m_{1}\times m_2 = -1$

Hence, a line perpendicular to given line will have slope:

$m_{1}\times m_2 = -1\\\displaystyle\frac{5}{2}\times m_2 = -1\\\\m_2 = \frac{-2}{5}$

Point slope form of a straight line:

$(y-y_1) = m(x-x_1)$

To find equation of line perpendicular to the given line and passing through the point (5,-4), we put the following in the above equation of line:

$m = \diplaystyle\frac{-2}{5}, (x_1, y_1) = (5,-4)$

Equation of line:

$y-(-4) = \displaystyle\frac{-2}{5}(x-5)\\\\5(y + 4) = -2(x-5)\\5y + 20 = -2x + 10\\2x + 5y = -10$

Option B) is the correct equation.

Answer 2

For this case we have that by definition, the equation of a line in the slope-intersection form is given by:

$y = mx + b$

Where:

m: It's the slope

b: It is the cut-off point with the y axis

On the other hand we have that if two lines are perpendicular, then the product of their slopes is -1. So:

$m_ {1} * m_ {2} = - 1$

The given line is:

$5x-2y = -6\\-2y = -6-5x\\2y = 5x + 6\\y = \frac {5} {2} x + \frac {6} {2}\\y = \frac {5} {2} x + 3$

So we have:

$m_ {1} = \frac {5} {2}$

We find $m_ {2}:$$m_ {2} = \frac {-1} {\frac {5} {2}}\\m = - \frac {2} {5}$

So, a line perpendicular to the one given is of the form:

$y = - \frac {2} {5} x + b$

We substitute the given point to find "b":

$-4 = - \frac {2} {5} (5) + b\\-4 = -2 + b\\-4 + 2 = b\\b = -2$

Finally we have:

$y = - \frac {2} {5} x-2$

In point-slope form we have:

$y - (- 4) = - \frac {2} {5} (x-5)\\y + 4 = - \frac {2} {5} (x-5)$

ANswer:

$y = - \frac {2} {5} x-2\\y + 4 = - \frac {2} {5} (x-5)$