How do you write an equation in point slope and slope intercept form given (8, -2) and perpendicular to the line whose equation is x-5y-7=0?

Question

Julia Adams · Accepted Answer

Point slope form equation: #(y+2)=-5(x-8)#
Slope intercept form equation: #y= -5 x +38# Slope of the line, # x-5 y -7=0 or y= 1/5 x -7/5; [y=m x+c]# is #m_1=1/5# [Compared with slope-intercept form of equation] The product of slopes of the pependicular lines is #m_1*m_2=-1# #:.m_2=(-1)/(1/5)=-5#. In point slope form, equation of line passing through #(x_1=8,y_1=-2)# having slope of #m_2=-5# is #y-y_1=m_2(x-x_1):. y-(-2)= -5 (x-8) #or #(y+2)=-5(x-8)#. Slope intercept form: #(y+2)=-5(x-8) or y +2 =-5 x+40 # or #y= -5 x +40-2 or y= -5 x +38# [Ans]

Eva Abramson · Answer

undefined To write the equation of a line in point-slope form and slope-intercept form, perpendicular to the line $x - 5y - 7 = 0$ and passing through the point (8, -2), follow these steps:

1. Convert the given equation $x - 5y - 7 = 0$ into slope-intercept form $y = mx + b$.
2. Find the slope of the given line.
3. Determine the slope of the line perpendicular to the given line.
4. Use the point-slope form $y - y_1 = m(x - x_1)$ with the given point (8, -2) and the perpendicular slope to find the equation in point-slope form.
5. Convert the equation from point-slope form to slope-intercept form.

Let's solve it step by step:

1. Convert the given equation $x - 5y - 7 = 0$ to slope-intercept form:

$$x - 5y - 7 = 0$$
$$x - 7 = 5y$$
$$y = \frac{1}{5}x - \frac{7}{5}$$

2. The slope of the given line is $m = \frac{1}{5}$.

3. The slope of the line perpendicular to the given line is the negative reciprocal of the slope of the given line. So, the perpendicular slope is $m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{5}{1} = -5$.

4. Using the point-slope form with the given point (8, -2) and the perpendicular slope:

$$y - (-2) = -5(x - 8)$$
$$y + 2 = -5x + 40$$

5. Convert the equation to slope-intercept form:

$$y = -5x + 40 - 2$$
$$y = -5x + 38$$

So, the equation of the line in both point-slope form and slope-intercept form, perpendicular to the line $x - 5y - 7 = 0$ and passing through the point (8, -2), is:

Point-slope form: $y + 2 = -5(x - 8)$

Slope-intercept form: $y = -5x + 38$